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Consecutive integer problems are word problems that involve consecutive integers.

**Consecutive integers** are integers that follow in sequence, each number being 1 more than the previous number, represented by *n*, *n* +1, *n* + 2, *n* + 3, ..., where n is any integer.

For example: 23, 24, 25, …

If we start with an even number and each number in the sequence is 2 more than the previous number then we will get consecutive even integers.

For example: 16,18, 20, …

If we start with an odd number and each number in the sequence is 2 more than the previous number then we will get consecutive odd integers.

For example: 33, 35, 37, …

We will look at some examples of consecutive integer problems, consecutive odd integer problems and consecutive even integer problems.

Example : Consecutive Integer Problem

The sum of the least and greatest of 3 consecutive integers is 60. What are the values of the 3 integers?

Solution:

Step 1: Assign variables :

Letx= least integer

x+ 1 = middle integer

x+ 2 = greatest integer

Translate sentence into an equation.

Sentence: The sum of the least and greatest is 60.

Rewrite sentence:

x +(x +2) = 60

Step 2: Solve the equation

Combine like terms

2x+ 2 = 60

Isolate variable *x*

2x= 58

Step 3: Check your answer

29 + 29 + 2 = 60

The question wants all the 3 consecutive numbers: 29, 30 and 31

Answer: The 3 consecutive numbers are 29, 30 and 31.

Example:

(1) The sum of 5 consecutive integers is 200, what is the smallest of the 5 integers?

(2) The sum of 7 odd integers is 217, what is the largest of the integers?

Example: The sum of three consecutive integers is 24. Find the integers.

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