How To Solve The Sum Of Three Consecutive Integers Problem

How do you solve math problems where the question asks to detail the three consecutive integer numbers whose sum is x ?

The problem can easily be solved if you apply algebra. As there are 3 consecutive integer numbers involved in this problem you could represent these three numbers as n , n+1 and n+2 .

What Does n+1 Mean?

In the context of this representation, it means the next number after n .

Therefore, the next consecutive integer number after the initial number can be represented as n+1 .

To represent the second consecutive number after the initial number would be n+2 .

You can represent any number of consecutive numbers using this simple algebraic technique.

Sum Of Consecutive Integers

To represent the consecutive numbers as a sum you would just apply the plus sign between each, like so:

n + (n+1) + (n+2)

To help show where the next and second consecutive numbers are in the formula above I’ve wrapped them in parentheses.

Therefore, to simplify this question you could add all the like terms and the numbers together.

This would provide a simplified algebraic expression like so:

3n + 3

Or, factorised:

3(n + 1)


To apply this technique to any word problem asking what the three consecutive digits would be if their sum is x :

3(n + 1) = x

You could further modify the formula above in terms of n as this is the first number needed to report your answer:

n = \frac{x}{3} - 1

To be clear this formula provides the initial number . To complete the answer you may need to provide the three integers.

Therefore, the answer needs to be the initial number n and the next integer n+1 , and the second consecutive number n+2 .

Let’s demonstrate how to answer this properly using an example to illustrate.


If the sum of three consecutive integers is 30, what are the three integers?



Use the formula written above where n was the subject:

n = \frac{x}{3} - 1

Substitute the result of 30 for x :

n = \frac{\fcolorbox{blue}{aqua}{\color{blue}{30}}}{3} - 1

Then perform the necessary operations:

n = \fcolorbox{blue}{aqua}{\color{blue}{10}} - 1
n = \fcolorbox{blue}{aqua}{\color{blue}{9}}

Now, remember the question asks for three consecutive integers. This formula has only provided the initial integer.

To complete the answer, you would need to list the three numbers: 9, 10 and 11.


To find the three consecutive integers whose sum is a number, use the formula to find the first of the three consecutive numbers:

n = \frac{x}{3} - 1

When you’ve substituted the total in for x , process the formula to arrive at the solution for n .

Once you have this solution add 1 to get the next consecutive integer and then add one more to find the second consecutive integer. You will then have your three consecutive numbers for your answer .

To check your answer simply add all three numbers up, and this should match the number in the question.

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