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What Do Consecutive Integers Represent In Math Problems?

consecutive integers: Colorful number blocks

What do consecutive integers mean, and how do you approach consecutive integer problems? Well, consecutive simply means "in a row." So, the term consecutive integers just means "numbers in a row."

1, 2, 3, 4, 5 are a series of five consecutive integers.

1, 3, 5, 7 are a series of consecutive odd integers.

-12, -11, -10, -9, -8 are a consecutive series of negative integers.

And so on!

The Sum of Consecutive Integers

You might find the phrase “consecutive integers” in a word problem. The first part of tackling word problems is to understand all the terms used.

If a problem asks you to work with consecutive odd numbers, for example, you need to first understand what "odd numbers" are. Then, you need to understand what "consecutive" means. Putting those together, it’s clear you are working with multiple odd numbers in a row, whether it is 1, 3, 5, 7, or another consecutive odd number series such as 133, 135, 137, 139.

You’ll often encounter a word problem that asks you for the sum of consecutive integers. To figure out what the sum of the integers equals, you need to know the first integer and the pattern for the consecutive integers.

For example, 5, 9, 13, 17, 21 is a series of consecutive integers that starts with 5 and increases by 4. Each number in the series is "consecutive" in that you can figure out the value for any of the next numbers by simply taking the previous number plus 4.

Usually, you will need algebra to figure this out. For example, if you know that the sum of the five consecutive odd numbers is 65, here’s how you would represent that and solve for x:

consecutive integers: Formula showing the sum of 5 consecutive odd numbers is 65

consecutive integers: Formula showing the sum of 5 consecutive odd numbers is 65

consecutive integers: Formula showing the sum of 5 consecutive odd numbers is 65

Formula showing the sum of five consecutive odd numbers is 65

Thus, you know the first term is equal to 9, and the five consecutive odd numbers equal 9, 11, 13, 15, 17. Let’s check by subbing in 9 for x:

Formula of 5 consecutive odd numbers with the sum of 65 by subbing in 9 for x

Formula of 5 consecutive odd numbers with the sum of 65 by subbing in 9 for x

Consecutive Prime Numbers

Consecutive prime numbers are trickier. First, remember that prime numbers are not divisible by any other number other than itself and 1. Consecutive prime numbers are a sequence of prime numbers such that there are no unlisted prime numbers in between each number.

For example, here’s a series of prime numbers: 2, 3, 5, 7, 11. Each of those numbers is prime, and there are no other possible prime numbers between each of those numbers. With consecutive prime numbers, the difference between the first integer to the second integer is not consistent throughout the rest of the sequence.

Solving Consecutive Integer Word Problems

As is good practice for any homework or test prep, read the problem carefully. Define the consecutive integers by finding the first term and then the pattern for the series. See if you can start with the first number and logically come up with each of the next numbers according to the pattern you find. Are they consecutive positive integers? Negative integers? Consecutive prime numbers?

With these steps, you easily work with consecutive integers in your math problems.

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