In these lessons, we will learn how to solve
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, …
Consecutive Integer Problems
Consecutive integer problems are word problems that involve consecutive integers.
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 equationCombine 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 31Answer: The 3 consecutive numbers are 29, 30 and 31.
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?
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 sidesStep 3: Plug in the values from the question and from the sketch.
45 = x + x + 2 + x + 4Combine like terms
45 = 3x + 6Isolate variable x
3x = 45 – 6Step 3: Check your answer
3x = 39
13 + 13 + 2 + 13 + 4 = 45
Be careful! The question requires the length of the longest side.
The length of longest = 13 + 4 =17Answer: The length of longest side is 17
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?
Step 1: Being consecutive even numbers we need to add 2 to the previous number.Assign variables :
Let x = length of first shelfStep 2: Convert 5 feet to inches
x + 2 = length of second shelf
x + 4 = length of third shelf
x + 6 = length of fourth shelf
5 × 12 = 60Step 3: Sum of the 4 shelves is 60
x + x + 2 + x + 4 + x + 6 = 60Combine like terms
4x + 12 = 60Isolate variable x
4x = 60 – 12Step 3: Check your answer
4x = 48
x = 12
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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