Algebra: Consecutive Integer Problems

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, …

The following are common examples of consecutive integer problems.

Example 1: 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 :

Let x = 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 2: Consecutive Odd Integer

The lengths of the sides of a triangle are consecutive odd numbers. What is the length of the longest side if the perimeter is 45?

Solution:

Step 1: Being consecutive odd numbers we need to add 2 to the previous number.

Assign variables :

Let x = length of shortest side

x + 2 = length of medium side

x + 4 = length of longest side

Sketch the figure

Step 2: Write out the formula for perimeter of triangle.

P = sum of the three sides

Step 3: Plug in the values from the question and from the sketch.

45 = x + x + 2 + x + 4

Combine like terms

45 = 3x + 6

Isolate variable x

3x = 45 – 6

3x = 39

x =13

Step 3: Check your answer

13 + 13 + 2 + 13 + 4 = 45

Be careful! The question requires the length of the longest side.

The length of longest = 13 + 4 =17

Answer: The length of longest side is 17

Example 3: Consecutive Even Integer

John has a board that is 5 feet long. He plans to use it to make 4 shelves whose lengths are to be a series of consecutive even numbers. How long should each shelf be in inches?

Solution:

Step 1: Being consecutive even numbers we need to add 2 to the previous number.

Assign variables :

Let x = length of first shelf

x + 2 = length of second shelf

x + 4 = length of third shelf

x + 6 = length of fourth shelf

Step 2: Convert 5 feet to inches

5 × 12 = 60

Step 3: Sum of the 4 shelves is 60

x + x + 2 + x + 4 + x + 6 = 60

Combine like terms

4x + 12 = 60

Isolate variable x

4x = 60 – 12

4x = 48

x = 12

Step 3: Check your answer

12 + 12 + 2 + 12 + 4 + 12 + 6 = 60

The lengths of the shelves should be 12, 14, 16 and 18.

Answer: The lengths of the shelves in inches should be 12, 14, 16 and 18.

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