Math/Consecutive Integers

## What are Consecutive Integers?

A consecutive integer is a number that follows each other in order. They have a difference of a certain number (a pattern). In a set of consecutive integers, the mean and the median are equal.

**For example**,
If x is an integer, then x, x+1, and x+2 would be consecutive integers.

For **in-dept** details, here are a few examples.

* Example 1.) *1, 2, 3, 4, 5*

## Even consecutive integers

Even consecutive integers have a difference of 2 between every two numbers.

If "x" is an even integer, then x, x+2, x+4 and x+6 will be **even consecutive integers**.

* Example 2.) *2, 4, 6, 8, 10, 12*

* Example 3.) *−2, 0, 2, 4, 6*

* Example 4.) *-6, -4, -2, 0*

## Odd consecutive integers

If "x" is an odd integer, then x, x+2, x+4 and x+6 will be **odd consecutive integers**. The first number in the pattern is always the variable

* Example 5.) *3, 5, 7, 9, 11*

* Example 6.) *−19, −17, −15, -13, -11*

## Example Questions

* Example Question 1
* Explanation
 Five consecutive integers are such that five times the smallest is 16 more than the largest. Find the integers.

So, we need to set up an equation but first, we need to label the **5 consecutive integers**. By doing this, we label x=**1st**, x+1=**2nd**, x+2=**3rd**, x+3=**4th**, and x+4=**5th**(For a clear view of this, look at the 5 consecutive integers to your right.

   Now, we need to set up an equation... By doing this we read the details on the problem **carefully**.
* "Such that **five** times the **smallest**" - So, we write down 5x because **five = 5** and **smallest = x**(x is the **smallest** because it is the first, look at your **right**.)
* "Is **16 more** than the **largest**" - On this, we write down **(x+4)** because it is the **largest** (look to your **right**). On the right of **(x+4)**, we put down **+16** because it says 16 more than the **largest**.
* Now you solve the problem, your equation should look like this:

* Now we need to solve this equation. We need to combine the like terms together so, you would need to combine **4** and **16** to get **20**.So it should look like **5x=x+20**.
* Now, you need to combine the **5x** and **x** to get **4x**. Now it should look like **4x=20**.
* Finally, you divide 4x on both sides, giving you **x=5**... Finished! It should look like this (on the right):

 \*Now you need to find the integers. Since we got **x=5**, we have to find the next five integers. Those would be **5, 6, 7, 8, 9**.

* We are now **done** with this question.

* Still confused?
* No worries, there are more problems!
* Scroll down and try to solve the next question below!
* Another alternative way (Not the best) is to look up this topic on YouTube!
* Math really shouldn't be confusing. It should fit in your brain like a puzzle.
* After understanding it, it'll be easy as pie!

* Example Question 2
* Explanation

Find two consecutive odd integers whose sum is -88.

Now, first we need to set up an equation in order to solve this problem. First off, we need to find the key words to help us build an equation.

* "Find **two** consecutive odd integers whose sum is -88". - For the two consecutive integers, we need to label down the integers down, **x=one x+2=two**.
* "Find two consecutive odd integers whose **sum is -88**". - So now we know that if we **add up the two consecutive odd integers**, it will be **-88 in total**.
* Now we have to set up an equation. By doing this, we can add up both of the consecutive integers because it **doesn't state specific ones**. So, the equation should look like this when combined:  **x+(x+2)=-88**
* Now, we solve the problem. **x** and **x** combined equals **2x**. So now, the equation should look like: **2x+2=-88**.
* We need to subract (opposite signs) **2** by **itself** and **-88**. Now it should look like this: **2x=-90**.
* We divide 2x on each side and we will end up with **x=-45** because **-90÷2** is **-45**.
* The steps should look like this (on your right)
* Now, we just need to find the two **odd** consecutive integers. Since we already got **-45** (counts as **one odd consecutive integer**), we just go on to the **next odd integer** which is **-43**. So, the two odd consecutive integers are **-45, -43**.
* Now we have **finished** solving question 2.

* Still confused?
* No worries, there are more problems!
* Scroll down and try to solve the next question below!
* Another alternative way (Not the best) is to look up this topic on YouTube!
* Math really shouldn't be confusing. It should fit in your brain like a puzzle.
* After understanding it, it'll be easy as pie!

* Example Question 3
* Explanation

Find three consecutive integers such that the sum of twice the smallest and 3 times the largest is 126.

* We need to label the three consecutive integers, that would be **x=1st**, **x+1=2nd**, **x+2=3rd**. For a more clear image, look to your right.
* Now we can set up an equation, but first we need to look at the key words from the question. It states "Find three consecutive integers such that the sum of **twice the smallest** and **3 times the largest** is 126".
* So, it says "**twice the smallest**" so lets take a look the the right. It says that x=1st (which is the smallest). So twice the smallest would be **2x**(because we are multiplying).
* It also says "**3 times the largest**". The largest is **x+2** (on the right) so it would be **3x+2**.
* So the sum of both of them would be **126**... Let's set up the equation!
* The equation would be **2x+3(x+2)=126**.
* Now, let's solve the equation. We first **distribute** the **3** to **x+2** so it would be **2x+3x+6=126**. Now, we combine the like terms, **2x** and **3x** to **5x**. So the equation should look like this now: **5x+6=126**. We now have to **subtract** the **6** from **itself** and from **126** so the equation would be **5x=120**... We just need to **divide 5 from both** which would be **x=24**. The steps should look like this. (on the right)
* Now, we just have to find the **3 consecutive integers**. So, **x=24** would be one, the next ones would be **25** and **26**. It should look like this: **24, 25, 26**.

* Still confused?
* No worries, there is more help available!
* Another alternative way (Not the best) is to look up this topic on YouTube!
* Math really shouldn't be confusing. It should fit in your brain like a puzzle.
* After understanding it, it'll be easy as pie!