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what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 .

for the recursive formula equation in your arithmetic example , do you have to state that 'k ' is greater than or equal to 2 ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here .

is there a way ( other than using a sequence ) to define a function limited to integers ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case .

is there a specific formula for recursive arithmetic equations ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ?

what is explicit and recursive ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers .

why is n't the infinite sequence just a ( k ) = 4k-1 instead of a ( k ) = 3+4 ( k-1 ) ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ?

are inductive definitions and explicit definitions the same ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

if you have an infinitely large sequence and you only want to take some of the sequence you could say n_k-1 since n_k is the `` end '' of the sequence right ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 .

does the ( k-1 ) have to be added in every sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers .

how the heck did sal figure out the equation for the finite sequence ( 1+3 ( k-1 ) ) ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers .

so one uses k-1 as to represent the nessecary integer previous to k ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 .

i got vey confused when sal started doing the '' a sub k = something ... '' can someone help me ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers .

testing function : a ( k ) = 1 + 3 ( k-1 ) , the numbers that work are replacing k with the position in the sequence , not any actual value ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 .

in every sequence would it be k-1 ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ?

which denotation of { a } is explicit and which is recursive ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so let 's see . when k is 3 , we added 3 twice . let me make it clear .

what are the multiples of 3 ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so let 's see . when k is 3 , we added 3 twice . let me make it clear .

( 3,6,9.. ) how much less from these multiples are the terms ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term .

if i wanted to write a more general equation of a ( sub ) k= ... , can i write a ( sub ) k=x+y ( k-1 ) ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences .

do we always have to put 12345 in the table or other numbers too ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence .

so can finite sequence be a irregular set of numbers ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 .

what is with the sub k ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 .

if not why do sum 's of series converge to a number ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

that 's not an attractive color . let me write this in . this is an explicit function .

how do i actually write an explicit formula ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 .

is it something like a^n=s+d ( n-1 ) , when s can be -5 and d is 2 ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 .

when you do the explicit equation why do you do k-1 or a -1 ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ?

what is the advantage of recursive functions over explicit ones ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with .

would n't the equations need to be simplified ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens .

when the sequence is not addition , but something more complex involving exponents and addition , how do you find the pattern ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing .

what does this man mean by we added 3 times one less than the k term times ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 .

do we have to use k when writing out our formulas ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ?

are arithmetic progression and arithmetic sequence the same ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 .

how does a sub number get the number on top ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence .

am i suppose to understand how sal wrote out the expression of the finite sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 .

throughout the entire video we use k-1 but would n't k-1 be zero and therefore not really exist since we began a k=1 since our initial term is defined at a_1=3 or a_1=1 in this video ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 .

what the point of writing k=1 ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

can a sequence a_n be undefined for some n ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ?

what is the diffrence between recursive and explicit formulae ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ?

sal what is the diffrence between a recursive and a explicit equation and how would you write a explicit rule from a recursive eqaution ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case .

we can see here that many of the simpler recursive sequences can be converted , but more generally , which kinds of recursive functions are convertible , and which , if any , are not ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term .

so what does the recursive formula look like generally ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case .

what is a recursive means ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 .

what if the sequence was not an arithmetic sequence for example 4 , -8,16 , -32,64 and i had to write a recursive formula for it ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ?

but what exactly are recursive and explicit forms .. ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity .

do all sequences use the variable k ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers .

i thought curly brackets were reserved for sets , where the order of elements is unimportant ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ?

so some can be classified as recursive and explicit ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence .

so to say it goes on forever can you put the infinity symbol ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times .

i know your'e given k=1 as a starting value but do you just ignore the function and write whatever the given starting value is ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 .

can you explain to me what ( k-1 ) means ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case .

what is the use of a recursive definition ... except for the fibonacci sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail .

do n't sequences have a connection to computer programming ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 .

for the infinite sequence when it shows { a ( sub k ) } ^infinity sub k=1 would it not be k=3 since the first term in the sequence is 3 ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case .

can a recursive equation include a negative beginning ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots .

how are arithmetic sequences and linear relationship related ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again .

can any of you explain to me the difference between a recursive and an explicit sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

is there such a thing called a geometric sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case .

what is the difference between implicit and recursive ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ?

could someone explain what explicit is ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 .

i do n't know where you get the k from and why you put a 4 on the corners of a.what are you basically doing afterwards ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ?

what is a mathematical , geometric , arithmetic , repeating , and growing patterns/sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 .

what is the sub k for ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here .

is there an easy way to define the function of any sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers .

is base k the same thing as an exponent ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences .

do you always add one to any number that you have ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

can we have a random sequence that the next element in the sequence is not predictable ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

does it matter what kind of brackets to draw ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 .

why do we need to specify that the lower limit of a sequence is 1 ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

also , would n't it be more simple to number the elements of a sequence starting with the number 0 ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

i know only arithmetic sequence and geometric sequence , so is infinite sequence and finite sequence are different with them ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so let 's see . when k is 3 , we added 3 twice . let me make it clear .

cant we write a sub k= k+3 ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out .

4 , how do you use a expression in a sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to .

are there different definitions for different type of sequences ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term .

what does sal mean when he says , `` a sub k '' ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 .

what is `` a sub k '' ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers .

if you are using k then what is a ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once .

how do you find the algebraic expression of a pattern that increases at different numbers each time ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ?

how would you figure out the recursive and explicit formulas if the pattern between the numbers increased each time ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

also , also how do you properly include a timestamp in your posts ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

what number is missing from the sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens .

: d and can you add a function to a sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 .

so for the first finite sequence a_k sal writes he really means to write a_k = 1 + 3 ( a_k-1 ) right ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

that 's not an attractive color . let me write this in . this is an explicit function .

what if its asking you to write an explicit or recursive rule for an exponent ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers .

what is the formula for pi ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 .

i understand the concept , however what is the definition of `` sub k '' ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 .

sal says `` a from 1 to 4 '' and writes a superscript `` 4 '' , but this is the number of terms -- and not the number at which the sequence ends -- correct ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

similarly , is it that the infinite sequence has an infinite number of terms , rather than `` going to infinity '' ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences .

is n't k also called n ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 .

is { ak } the set formula for sequences ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ?

is there a certain format for explicit and recursive formulas ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers .

how do you figure out the equations , such as 3 ( k-1 ) +1 ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time .

why do n't you use an= a1 + ( n-1 ) d. where an is the same as ak , a1 is the first term , n is any term ( which you usally solve for ) , and d is common difference ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

how do i graph a sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

what is the difference between sequence and series in maths ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

what if the sequence is decreasing ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with .

when would you want to use the recursive formula and when would you want to use the explicit ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 .

ok so if i have 2a^n-1 - n would i be multiplying the n-1 by the 2a ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers .

do sequences just go onto as far as the creator of that sequence want it to be ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 .

given the formula a ( sub n ) n=a ( sub n-1 ) +n^2 , would this formula be recursive because it references a previous term or explicit because it involves substituting your term value into the equation ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers .

should k=1 be k=3 instead ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 .

where the first term is 3 ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence .

does a finite sequence have to have a pattern ?