This lesson defines consecutive integers and gives fun examples of consecutive numbers in use. You will also learn the formulas used to compute consecutive integers.

## What is a Consecutive Integer?

An integer is simply a number like 0, 1, 2, 3, and 4, but unlike whole numbers, integers also include negative numbers like -1, -2, -3 and -4. An integer cannot be a decimal or a fraction.**Consecutive integers** are simply integers that follow each other in a patterned order, usually just one number after the other, like 1, 2, 3 and 4!

## Examples of Consecutive Integers

*Example 1*:Let’s imagine we are counting down to the new year!

The countdown to 1 when we are about to welcome a new year is an example of consecutive integers. They follow each other in order, just backwards.*Example 2*:Let’s imagine a teacher is counting the heads of students as they load the bus for a field trip.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12!The teacher has just counted all 12 children. Numbers 1-12 represent consecutive integers!

## Negative Consecutive Integers

We cannot forget that consecutive integers also include negative numbers.Take a look at this number line indicating consecutive numbers from -3 to 3:

## Even and Odd Consecutive Integers

**Even consecutive integers** are even numbers that follow each other in order.

The easiest example would be 2, 4, 6, 8 and 10.246, 248, 250 and 252 is another example of even consecutive integers.**Odd consecutive integers** are odd numbers that follow each other in order.The easiest example would be 1, 3, 5, 7 and 9.157, 159, 161 and 163 is another example of odd consecutive integers.

## Examples of Use of Even and Odd Consecutive Integers

*Example 1 (even)*:Ted, a middle school coach, wants to make two fair teams from his physical education class to play an indoor soccer game. He asks the class to line up and taps every other person, saying ‘2, 4, 6, 8’ etc. until the 20th person to make team A. He announces that the students that he did not tap (the students in odd position) will make up team B.So, Ted has used even consecutive integers (2, 4, 6, 8, 10, 12, 14, 16, 18 and 20) to make teams for the soccer game.

*Example 2 (odd)*:Mira has 30 gift bags for her 10th birthday pool party. She has 15 boys and 15 girls coming to her party. Her mother has lined up the bags so that they alternate between boy and girl. Mira’s mother has already filled the girl bags with goodies.

She asks Mira for help in filling the boy bags. Mira’s mother asks her to only fill the bags from the very first bag (#1) to the second to last bag (#29) using odd consecutive integers.Mira then fills bags 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 and 29 for the boy gift bags.So, Mira has used odd consecutive integers to fill up the party bags.

## Formula for Consecutive Integers

The formula for consecutive integers is pretty straightforward.

If x is the first consecutive integer, then x+1 will be the second, x+2 will be the third, x+3 will be the fourth, and so on.Let’s look at an example.If 45 is the first consecutive integer in a series of numbers, 45 would be ‘x.

‘Therefore x+1 (45 +1) will be the second consecutive number in the series. 45 + 1 = 46x+2 will be the third consecutive number in this series. 45 + 2 = 47If we are trying to figure out consecutive even or odd integers, the formula is a little different.If x is the first even or odd consecutive integer, x+2 will be the second, x+4 will be the third, x+6 will be the fourth, and so on.If trying to find the value of ‘x’ or the sum of a series of consecutive numbers, the formula would look like this:(x) + (x+1) = Sum of the Consecutive IntegersDepending on how many integers are in a set, you continue adding + (x+2) + (x+3) + (x+4) and so on.Example:There are three consecutive integers and the sum is 18.

What is the value of x (the first consecutive integer)?Let’s solve the above problem:(x) + (x+1) + (x+2) = 18x + x + 1 + x + 2= 183x + 3 = 183x = 18 – 33x = 15x = 5Let’s check our result. If x=5, that means the first consecutive integer out of three is 5.Does 5+6+7= 18? Yes! The formula worked!

## Lesson Summary

Let’s review:Integers are simply odd and even numbers, including zero and negative numbers, but not including fractions or decimals.

**Consecutive integers** are simply integers that follow each other. We can have **even consecutive integers** and **odd consecutive integers**, as well.The formula to find consecutive integers is:x, x+1, x+2, x+3, x+4, etc. depending on how many consecutive integers you want to find.

The formula to find even or odd consecutive integers is:x, x+2, x+4, x+6, x+8, etc. depending on how many even or odd consecutive integers you want to find.Remember, if asked to find the value of x (the first consecutive integer in a set) or the sum, the formula will look like this:(x) + (x+1) = Sum of the Consecutive IntegersDepending on how many integers are in a set, you continue adding + (x+2) + (x+3) + (x+4) etc.