Techniques for Adding the Numbers 1 to 100 – BetterExplained 8 Other Posts In This Series Technique 1: Pair Numbers Pairing numbers is a common approach to this problem. Instead of writing all the numbers in a single column, let’s wrap the numbers around, like this: 1 2 3 4 5 10 9 8 7 6 An interesting pattern emerges: the sum of each column is 11. As the top row increases, the bottom row decreases, so the sum stays the same. Because 1 is paired with 10 (our n), we can say that each column has (n+1). And how many pairs do we have? Well, we have 2 equal rows, we must have n/2 pairs. which is the formula above. Wait — what about an odd number of items? Ah, I’m glad you brought it up. What if we are adding up the numbers 1 to 9? We don’t have an even number of items to pair up. Many explanations will just give the explanation above and leave it at that. I won’t.

Let’s add the numbers 1 to 9, but instead of starting from 1, let’s count from 0 instead: 0 1 2 3 4 9 8 7 6 5 By counting from 0, we get an “extra item” (10 in total) so we can have an even number of rows. However, our formula will look a bit different. Notice that each column has a sum of n (not n+1, like before), since 0 and 9 are grouped. And instead of having exactly n items in 2 rows (for n/2 pairs total), we have n + 1 items in 2 rows (for (n + 1)/2 pairs total). If you plug these numbers in you get: which is the same formula as before. It always bugged me that the same formula worked for both odd and even numbers – won’t you get a fraction? Yep, you get the same formula, but for different reasons. Technique 2: Use Two Rows The above method works, but you handle odd and even numbers differently. Isn’t there a better way? Yes.

Instead of looping the numbers around, let’s write them in two rows: 1 2 3 4 5 6 7 8 9 10 10 9 8 7 6 5 4 3 2 1 Notice that we have 10 pairs, and each pair adds up to 10+1. The total of all the numbers above is But we only want the sum of one row, not both. So we divide the formula above by 2 and get: Now this is cool (as cool as rows of numbers can be). It works for an odd or even number of items the same! Technique 3: Make a Rectangle I recently stumbled upon another explanation, a fresh approach to the old pairing explanation. Different explanations work better for different people, and I tend to like this one better. Instead of writing out numbers, pretend we have beans. We want to add 1 bean to 2 beans to 3 beans… all the way up to 5 beans. x x x x x x x x x x x x x x x Sure, we could go to 10 or 100 beans, but with 5 you get the idea.

How do we count the number of beans in our pyramid? Well, the sum is clearly 1 + 2 + 3 + 4 + 5. But let’s look at it a different way. Let’s say we mirror our pyramid (I’ll use “o” for the mirrored beans), and then topple it over: x o x o o o o o x x o o x x o o o o x x x o o o => x x x o o o x x x x o o o o x x x x o o x x x x x o o o o o x x x x x o Cool, huh? In case you’re wondering whether it “really” lines up, it does. Take a look at the bottom row of the regular pyramid, with 5′x (and 1 o). The next row of the pyramid has 1 less x (4 total) and 1 more o (2 total) to fill the gap. Just like the pairing, one side is increasing, and the other is decreasing. Now for the explanation: How many beans do we have total? Well, that’s just the area of the rectangle.

We have n rows (we didn’t change the number of rows in the pyramid), and our collection is (n + 1) units wide, since 1 “o” is paired up with all the “x”s. Notice that this time, we don’t care about n being odd or even – the total area formula works out just fine. If n is odd, we’ll have an even number of items (n+1) in each row. But of course, we don’t want the total area (the number of x’s and o’s), we just want the number of x’s. Since we doubled the x’s to get the o’s, the x’s by themselves are just half of the total area: And we’re back to our original formula. Again, the number of x’s in the pyramid = 1 + 2 + 3 + 4 + 5, or the sum from 1 to n. Technique 4: Average it out We all know that average = sum / number of items which we can rewrite to sum = average * number of items So let’s figure out the sum.

If we have 100 numbers (1…100), then we clearly have 100 items. That was easy. To get the average, notice that the numbers are all equally distributed. For every big number, there’s a small number on the other end. Let’s look at a small set: 1 2 3 The average is 2. 2 is already in the middle, and 1 and 3 “cancel out” so their average is 2. For an even number of items 1 2 3 4 the average is between 2 and 3 – it’s 2.5. Even though we have a fractional average, this is ok — since we have an even number of items, when we multiply the average by the count that ugly fraction will disappear. Notice in both cases, 1 is on one side of the average and N is equally far away on the other. So, we can say the average of the entire set is actually just the average of 1 and n: (1 + n)/2. Putting this into our formula And voila! We have a fourth way of thinking about our formula. So why is this useful?

Three reasons: 1) Adding up numbers quickly can be useful for estimation. Notice that the formula expands to this: Let’s say you want to add the numbers from 1 to 1000: suppose you get 1 additional visitor to your site each day – how many total visitors will you have after 1000 days? Since thousand squared = 1 million , we get million / 2 + 1000/2 = 500,500. 2) This concept of adding numbers 1 to N shows up in other places, like figuring out the probability for the birthday paradox . Having a firm grasp of this formula will help your understanding in many areas. 3) Most importantly, this example shows there are many ways to understand a formula. Maybe you like the pairing method, maybe you prefer the rectangle technique, or maybe there’s another explanation that works for you. Don’t give up when you don’t understand — try to find another explanation that works. Happy math. By the way, there are more details about the history of this story and the technique Gauss may have used. Variations Instead of 1 to n, how about 5 to n?

Start with the regular formula (1 + 2 + 3 + … + n = n * (n + 1) / 2) and subtract off the part you don’t want (1 + 2 + 3 + 4 = 4 * (4 + 1) / 2 = 10). Sum for 5 + 6 + 7 + 8 + … n = [n * (n + 1) / 2] – 10 And for any starting number a: Sum from a to n = [n * (n + 1) / 2] – [(a - 1) * a / 2] We want to get rid of every number from 1 up to a – 1. How about even numbers, like 2 + 4 + 6 + 8 + … + n? Just double the regular formula.

To add evens from 2 to 50, find 1 + 2 + 3 + 4 … + 25 and double it: Sum of 2 + 4 + 6 + … + n = 2 * (1 + 2 + 3 + … + n/2) = 2 * n/2 * (n/2 + 1) / 2 = n/2 * (n/2 + 1) So, to get the evens from 2 to 50 you’d do 25 * (25 + 1) = 650 How about odd numbers, like 1 + 3 + 5 + 7 + … + n? That’s the same as the even formula, except each number is 1 less than its counterpart (we have 1 instead of 2, 3 instead of 4, and so on).

We get the next biggest even number (n + 1) and take off the extra (n + 1)/2 “-1″ items: Sum of 1 + 3 + 5 + 7 + … + n = [(n + 1)/2 * ((n + 1)/2 + 1)] – [(n + 1) / 2] To add 1 + 3 + 5 + … 13, get the next biggest even (n + 1 = 14) and do [14/2 * (14/2 + 1)] – 7 = 7 * 8 – 7 = 56 – 7 = 49 Combinations: evens and offset Let’s say you want the evens from 50 + 52 + 54 + 56 + … 100.

Find all the evens 2 + 4 + 6 + … + 100 = 50 * 51 and subtract off the ones you don’t want 2 + 4 + 6 + … 48 = 24 * 25 So, the sum from 50 + 52 + … 100 = (50 * 51) – (24 * 25) = 1950 Phew! Hope this helps. Ruby nerds: you can check this using (50..100).select {|x| x % 2 == 0 }.inject(:+) 1950 Mr. Average - The true story about penis size, from a site that isn't trying to sell you anything. The quick answer is that the average length is somewhat under 6" and the circumference is just under 5". Or to be more exact, the average length seems to be between 5.65" and 5.87". The average circumference between 4.67" and 4.97". Both depending on how you interpret the data. If you really want to go into detail, just keep reading.

Here is the only survey that I've found that was anything like scientific, so probably gives as accurate an average as you are going to get. Even with this the raw data is not available, so I've just done the best I can with it. have conducted research into the average length and circumference of the erect penis during Spring Break in Cancun and found that the average penis length is 5.877 inches, with the majority ranging between 5.5 inches and 6.3 inches. The average girth is 4.972 inches, with the majority ranging between 4.7 inches and 5.1 inches. From March 11 to 17, a total 401 males over the age of 18 were measured, one at a time, by qualified medical staff (protected by gloves) in private tents at the Dady Rock nightclub in Cancun, Mexico. Of the 401 men, 300 were able to gain an erection for measurement  a success rate of only 75%. Measurement methodology The following criteria were enforced by a doctor and a team of four nurses, who attended each of the seven nights of the penis size research.

Measuring of length was from the point at the base of the erect penis where the top of the penile shaft meets the pubic area to the tip of the glans ("head"). For uncircumcised men, the foreskin was retracted. Measuring of girth was at mid-shaft around the circumference of the penis when the penis was erect. Failure to gain an effective erection for measurement disqualified the volunteer from being included in the data for the calculation of average erect penis length and girth. Environment The majority of participants were U.S. college men, but it was not limited to this group. Volunteers were asked to enter a private area, where they would gain an erection and then call for the two nurses to measure their erect penis. ( Male fantasy overload.) To ensure the accuracy of the data, each erect penis was measured by two of the nurses. A doctor supervised the nurses and all activity. Volunteers tended to be groups of friends, which helped ensure a wide range of penis sizes (big and small) and provide a more accurate average figure.

Sample size The sample included 401 male volunteers, of which only 300 were "effective" male volunteers (75 percent), because 101 volunteers (25 percent) could not gain an erection for measurement. At 300 effective measurements, the study was twice the size of any previous study that used medical staff to measure penis size. This method is more accurate and reliable than studies in which volunteers undertake self-measurement/reporting. Yes I do know that these statistics have flaws, but it's the best I could find, without using any self reported measurements. There is a more in depth analysis of the results here . Erect penis length The average erect penis length was 5.877 inches (14.928 cm). Standard deviation was 0.825 inches (2.096 cm) with 54 percent of the effective sample measuring between 5.5 inches (14 cm) and 6.3 inches (16 cm). Length Arithmetic Mean Arithmetic Mean Unit 8 > Lesson 2 of 8 Problem: Scott took 7 math tests in one marking period. What is the mean test score?

89, 73, 84, 91, 87, 77, 94 Solution: The sum of these numbers is 595. Dividing the sum by the number of test scores we get: The mean test score is 85. Definition: The arithmetic mean of a set of data is found by taking the sum of the data, and then dividing the sum by the total number of values in the set. A mean is commonly referred to as an average. In the problem above, the mean was a whole number. This is not always the case. Let's look at some more examples. Example 1: Find the mean driving speed for 6 different cars on the same highway. 66 mph, 57 mph, 71 mph, 54 mph, 69 mph, 58 mph Solution: 66 + 57 + 71 + 54 + 69 + 58 = 375 Answer: The mean driving speed is 62.5 mph. Example 2: The Scheuern family drove through 4 midwestern states on their summer vacation. Gasoline prices varied from state to state.

What is the mean gasoline price? $1.79 + $1.61 + $1.96 + $2.08 = $7.44 The mean gasoline price is $1.86. Example 3: A marathon race was completed by 5 participants in the times given below. What is the mean race time for this marathon? 2.7 hr, 8.3 hr, 3.5 hr, 5.1 hr, 4.9 hr Solution: 2.7 + 8.3 + 3.5 + 5.1 + 4.9 = 24.5 The mean race time is 4.9 hr Example 4: Find the mean swimming time rounded to the nearest tenth: 2.6 min, 7.2 min, 3.5 min, 9.8 min, 2.5 min Solution: 2.6 + 7.2 + 3.5 + 9.8 + 2.5 = 25.6 The mean swimming time to the nearest tenth is 5.1 min.

Summary: The arithmetic mean of a set of n numbers is the sum of the n numbers divided by n. The mean is commonly referred to as an average. Exercises Directions: Find the mean of each set of data. Click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR. 1. AlgebraLAB: Lessons Finding the Mean, Median, and Mode When you hear someone use the word "average" to describe a set of numbers, be careful. A more accurate phrase to use is measure of central tendency. There are three ways you can describe data, all of which can be thought of as average. The mean of a set of data is probably what most people refer to as average. You find the mean of a set of data by adding up all the numbers in the data set and then dividing by the number of data points you added up. When someone says average, this is usually what they are talking about. However, there is another number that is often used to describe a set of data which is the median.

If you think about driving along an interstate highway, you often see signs that say “Keep Off Median” meaning that you can’t drive on the grassy area in the middle of the interstate. Median in statistics is the same thing. Well, it’s not a grassy area, but the median does refer to the number in the middle of a data set. When finding the median of a data set, you have to make sure that the numbers are put in order first. If there is an odd number of data in the list, there is only one number that is exactly in the middle of the data. But if there is an even number of data points, then there are two numbers in the middle. In that case, you have to add those two numbers together and then divide by two to find the median. The mode of a data set refers to the number that occurs most often. If there is not a number that occurs more than any other, we say there is no mode for the data. It is possible to have more than one mode for a data set. Why in the world would there be three different ways to describe a set of data? It all depends on the situation. Generally speaking, statisticians tend to follow some simple rules.

If there are extremely high or extremely low values in the data set, those numbers can greatly affect the mean. In those cases of extreme values (in one direction or another) it is preferable to use the median because it is not affected by extreme values. When you have categorical data, or data that appears as words instead of numbers, you need to use the mode. For example, if I ask you your favorite color and get word answers like red, blue, green, purple, it is impossible to add up those values to find the mean or to order the colors and find a median. The only alternative to describe this set of data would be the mode . Then you can say that most people prefer red (or whatever color occurs most). You probably will not be faced with having to make a decision about which measure of central tendency will be best for a particular data set. Most books will give you a set of numbers and ask you to compute mean, median, and mode. But many students often wonder why there are three different values to be found from the same data set. This brief explanation gives you an idea of the different types of uses, just in case you were wondering.

Let's Practice: Given the following data set, find the mean, median, and mode. 12, 15, 16, 19, 20, 20, 22, 23, 25, 27, 29, 30, 32, 32, 35 The mean is found by adding the 15 numbers together which gives 357. Now take 357 and divide it by 15. This gives the value for the mean to be 23.8. To find the median, we have to make sure that the data set is in order. This data is already in order so we can proceed directly to finding the middle value. In a data set with 15 numbers, the median will be the 8th value. The median for this data set is 23. The mode is the value that occurs most often. There are two modes in this data set. The values 20 and 32 both appear twice meaning that both are considered the mode .

A survey of 20 students was conducted to find out how many books they had read during the past three months (including books for school). The results from those 20 students are shown below. Find the mean, median, and mode for this data. 2, 4, 5, 1, 3, 2, 5, 6, 1, 2, 4, 3, 6, 10, 12, 10, 2, 8, 6, 7 Find the mean by adding all the numbers and dividing by 20. Adding all the numbers results in 99. Divide 99 by 20 and you get a mean of 4.95. Before finding the median, the numbers have to be put in order from smallest to largest.

The ordered data set is 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 8, 10, 10, 12 Now the median can be found by finding the middle value. In this case, there are 20 numbers in the data set which means there are two numbers in the middle. Those numbers are 4 and 5. The median is found by adding 4 and 5 and then dividing by 2. This gives a median value of 4.5. The mode is the value that occurs most often which in this set is 4. Examples Find the mean, median, and mode for the data set 5, 12, 7, 14, 8, 9, 4 What is your answer? Find the mean, median and mode for the data set 20, 39, 31, 37, 32, 33, 34, 32 What is your answer?

What happens to the mean, median, and mode of the data in problem #2 if you add 10 to each number? What is your answer? Based on what you found out in #3, what do you think would happen if you added 20 to each value in data set #1? What is your answer? Videos R Us keeps a record of how many movies are rented on each day of the week. Mon SOLUTION: Use the arithmetic sequence of numbers 1, 3, 5, 7, 9,�to find the following: a) What is d, the difference between any two consecutive terms? Answer: 2 Show work in this space. You can put this solution on YOUR website! a) The difference is the factor between each term. So going from 1 to 3, 3 to 5, 5 to 7, you see that its adding 2 each time. To verify, pick one term and subtract the previous term from it. So lets say I choose 7: I'm going to subtract 5 from it to get a difference of 2.

If I pick 5, and subtract 3, I get a difference of 2.So the difference is: d=2 b) Using what we found earlier, I know that the sequence counts up by 2 each term. So if I'm at 1 (the 1st term) and I go to 3, this means I increase by 2 each term. If I let n=0 then the term is 1, and if I let n=1 then the term is 3. This basically tells me that the arithmetic sequence is 2n+1. To verify, simply plug in the 1st term (n=0) and you'll get 1. Plug in the 2nd term (n=1) you'll get 3, if I let n=2 I get 5, etc.

If I wanted to know the 101st term, let n=100 (zero is the first term) and it comes to So the 101st term is 201 c) Using the sum of arithmetic series formula: a[1]=first term, a[n]=nth term (ending term which is the 20th term), and n is the number of terms Plug in values So the sum of the first 20 terms is 400.

d) Again using the same formula a[1]=first term, a[n]=nth term (ending term which is the 30th term), and n is the number of terms Plug in values Sum of the first 2 terms 1+3=4 Sum of the first 3 terms 1+3+5=9 Sum of the first 4 terms 1+3+5+7=16 Sum of the first 5 terms 1+3+5+7+9=25 Sum of the first 6 terms 1+3+5+7+9+11=36 Sum of the first 7 terms 1+3+5+7+9+11+13=49 Sum of the first 8 terms 1+3+5+7+9+11+13+15=64 Sum of the first 9 terms 1+3+5+7+9+11+13+15+17=81 Sum of the first 10 terms 1+3+5+7+9+11+13+15+17+19=100 Notice how

 the partial sums are all perfect squares.

So the sums follow the sequence -------------------------------------------------------------------------- Standard Deviation Formulas Standard Deviation Formulas Deviation just means how far from the normal Standard Deviation You might like to read this simpler page on Standard Deviation first. But here we explain the formulas. The symbol for Standard Deviation is σ (the Greek letter sigma). This is the formula for Standard Deviation: Say what? Please explain! OK. Let us explain it step by step. Say we have a bunch of numbers like 9, 2, 5, 4, 12, 7, 8, 11. To calculate the standard deviation of those numbers: 1. Work out the Mean (the simple average of the numbers) 2. Then for each number: subtract the Mean and square the result 3. Then work out the mean of those squared differences. 4. Take the square root of that and we are done! The formula actually says all of that, and I will show you how. The Formula Explained First, let us have some example values to work on: Example: Sam has 20 Rose Bushes.

The number of flowers on each bush is 9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4 Work out the Standard Deviation. Step 1. Work out the mean In the formula above μ (the greek letter "mu") is the mean of all our values ... Example: 9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4 The mean is: 9+2+5+4+12+7+8+11+9+3+7+4+12+5+4+10+9+6+9+4 20 = 140 20 = 7 ... etc ...

And we get these results: 4, 25, 4, 9, 25, 0, 1, 16, 4, 16, 0, 9, 25, 4, 9, 9, 4, 1, 4, 9 Step 3. Then work out the mean of those squared differences. To work out the mean, add up all the values then divide by how many. First add up all the values from the previous step. But how do we say "add them all up" in mathematics? We use "Sigma": Σ The handy Sigma Notation says to sum up as many terms as we want: Sigma Notation We want to add up all the values from 1 to N, where N=20 in our case because there are 20 values: Example (continued): Which means: Sum all values from (x1-7)2 to (xN-7)2 We already calculated (x1-7)2=4 etc.

in the previous step, so just sum them up: = 4+25+4+9+25+0+1+16+4+16+0+9+25+4+9+9+4+1+4+9 = 178 But that isn't the mean yet, we need to divide by how many, which is simply done by multiplying by "1/N": Example (continued): Mean of squared differences = (1/20) × 178 = 8.9 (Note: this value is called the "Variance") Step 4. Take the square root of that: Example (concluded): But wait, there is more ... ... sometimes our data is only a sample of the whole population. Example: Sam has 20 rose bushes, but only counted the flowers on 6 of them! The "population" is all 20 rose bushes, and the "sample" is the 6 bushes that Sam counted the flowers of. Let us say Sam's flower counts are: 9, 2, 5, 4, 12, 7 We can still estimate the Standard Deviation.

But when we use the sample as an estimate of the whole population, the Standard Deviation formula changes to this: The formula for Sample Standard Deviation: The important change is "N-1" instead of "N" (which is called "Bessel's correction"). The symbols also change to reflect that we are working on a sample instead of the whole population: The mean is now x (for sample mean) instead of μ (the population mean), And the answer is s (for Sample Standard Deviation) instead of σ. But that does not affect the calculations. Only N-1 instead of N changes the calculations. OK, let us now calculate the Sample Standard Deviation: Step 1. Work out the mean Example 2: Using sampled values 9, 2, 5, 4, 12, 7 The mean is (9+2+5+4+12+7) / 6 = 39/6 = 6.5 So: When we used the whole population we got: Mean = 7, Standard Deviation = 2.983...

When we used the sample we got: Sample Mean = 6.5, Sample Standard Deviation = 3.619... Our Sample Mean was wrong by 7%, and our Sample Standard Deviation was wrong by 21%. Why Take a Sample? Mostly because it is easier and cheaper. Imagine you want to know what the whole country thinks ... you can't ask millions of people, so instead you ask maybe 1,000 people. There is a nice quote (possibly by Samuel Johnson): "You don't have to eat the whole ox to know that the meat is tough." This is the essential idea of sampling. To find out information about the population (such as mean and standard deviation), we do not need to look at all members of the population; we only need a sample. But when we take a sample, we lose some accuracy. Sequences Sequences You can read a gentle introduction to Sequences in Common Number Patterns . What is a Sequence? A Sequence is a list of things (usually numbers) that are in order.

Infinite or Finite When the sequence goes on forever it is called an infinite sequence, otherwise it is a finite sequence Examples: {1, 2, 3, 4, ...} is a very simple sequence (and it is an infinite sequence) {20, 25, 30, 35, ...} is also an infinite sequence {1, 3, 5, 7} is the sequence of the first 4 odd numbers (and is a finite sequence) {4, 3, 2, 1} is 4 to 1 backwards {1, 2, 4, 8, 16, 32, ...} is an infinite sequence where every term doubles {a, b, c, d, e} is the sequence of the first 5 letters alphabetically {f, r, e, d} is the sequence of letters in the name "fred" {0, 1, 0, 1, 0, 1, ...} is the sequence of alternating 0s and 1s (yes they are in order, it is an alternating order in this case) In Order When we say the terms are "in order", we

 are free to define what order that is!

They could go forwards, backwards ... or they could alternate ... or any type of order we want! Like a Set A Sequence is like a Set , except: the terms are in order (with Sets the order does not matter) the same value can appear many times (only once in Sets) Example: {0, 1, 0, 1, 0, 1, ...} is the sequence of alternating 0s and 1s. The set is just {0,1} Notation The curly brackets { } are sometimes called "set brackets" or "braces". A Rule A Sequence usually has a Rule, which is a way to find the value of each term. Example: the sequence {3, 5, 7, 9, ...} starts at 3 and jumps 2 every time: As a Formula Saying "starts at 3 and jumps 2 every time" is fine, but it doesn't help us calculate the: 10th term, nth term, where n could be any term number we want. So, we want a formula with "n" in it (where n is any term number).

So, What Can A Rule For {3, 5, 7, 9, ...} Be? Firstly, we can see the sequence goes up 2 every time, so we can guess that a Rule is something like "2 times n" (where "n" is the term number). Let's test it out: Test Rule: 2n 2n+1 = 2×3 + 1 = 7 That Works! So instead of saying "starts at 3 and jumps 2 every time" we write this: 2n+1 Now we can calculate, for example, the 100th term: 2 × 100 + 1 = 201 Many Rules But mathematics is so powerful we can find more than one Rule that works for any sequence. Example: the sequence {3, 5, 7, 9, ...} We have just shown a Rule for {3, 5, 7, 9, ...} is: 2n+1 And so we get: {3, 5, 7, 9, 11, 13, ...} But can we find another rule?

How about "odd numbers without a 1 in them": And we get: {3, 5, 7, 9, 23, 25, ...} A completely different sequence! And we could find more rules that match {3, 5, 7, 9, ...}. Really we could. So it is best to say "A Rule" rather then "The Rule" (unless we know it is the right Rule). Notation To make it easier to use rules, we often use this special style: xn is the term n is the term number Example: to mention the "5th term" we write: x5 So a rule for {3, 5, 7, 9, ...} can be written as an equation like this: xn = 2n+1 And to calculate the 10th term we can write: x10 = 2n+1 = 2×10+1 = 21 Can you calculate x50 (the 50th term) doing this? Here is another example: {an} = { -1, 1/4, -1/27, 1/256, ...

} Special Sequences Now let's look at some special sequences, and their rules. Arithmetic Sequences In an Arithmetic Sequence the difference between one term and the next is a constant. In other words, we just add some value each time ... on to infinity. Example: 1, 4, 7, 10, 13, 16, 19, 22, 25, ... This sequence has a difference of 3 between each number. Its Rule is xn = 3n-2 In General we can write an arithmetic sequence like this: {a, a+d, a+2d, a+3d, ... } where: a is the first term, and d is the difference between the terms (called the "common difference") And we can make the rule: xn = a + d(n-1) (We use "n-1" because d is not used in the 1st term). Geometric Sequences In a Geometric Sequence each term is found by multiplying the previous term by a constant. Example: This sequence has a factor of 2 between each number.

Its Rule is xn = 2n In General we can write a geometric sequence like this: {a, ar, ar2, ar3, ... } a is the first term, and r is the factor between the terms (called the "common ratio") Note: r should not be 0. When r=0, we get the sequence {a,0,0,...} which is not geometric And the rule is: (We use "n-1" because ar0 is the 1st term) Triangular Numbers 1, 3, 6, 10, 15, 21, 28, 36, 45, ... The Triangular Number Sequence is generated from a pattern of dots which form a triangle.

By adding another row of dots and counting all the dots we can find the next number of the sequence: But it is easier to use this Rule: xn = n(n+1)/2 the 5th Triangular Number is x5 = 5(5+1)/2 = 15, and the sixth is x6 = 6(6+1)/2 = 21 Square Numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, ... The next number is made by squaring where it is in the pattern. Rule is xn = n2 Example: term "6" is calculated like this: x6 = x6-1 + x6-2 = x5 + x4 = 5 + 3 = 8 Series and Partial Sums Now you know about sequences, the next thing to learn about is how to sum them up. Read our page on Partial Sums . When we sum up just part of a sequence it is called a Partial Sum . But a sum of an infinite sequence it is called a "Series" (it sounds like another name for sequence, but it is actually a sum).

See Infinite Series . Example: Odd numbers Sequence: {1, 3, 5, 7, ...} Series: 1 + 3 + 5 + 7 + ... Partial Sum of first 3 terms: 1 + 3 + 5

Question: What is the average of 1, 3, 5, 6, 7, 9 and 25?
Answer: 8