[Maths Class Notes] on Consecutive Integers Pdf for Exam

When you start counting natural numbers you are just counting the consecutive numbers or consecutive integers. Consecutive integers are integers that follow each other in a fixed sequence. Did you know that whenever you number items you are using Consecutive Integers? In fact, whenever you count by ones from any number in a set you obtain Consecutive Integers. Consecutive integers are integers that follow in a fixed sequence, each number being 1 more than the previous number, Consecutive integers are represented by n, n +1, n + 2, n + 3, …, where n is an integer.

Examples of Consecutive Integer

For example: 23, 24, 25

Look at the following. The first set is called consecutive positive integers and the second set is called consecutive negative integers.

Example 1: 1, 2, 3, 4, 5…..

Example 2: -1, -2, -3, -4, -5, -6,…..

In the first example a set of consecutive integers is found by adding 1 to 0. You can represent the first set with this expression: n + 1, with n = 0, 1, 2, …..

The second set of consecutive integers is found by subtracting 1 from 0. You can represent the second set with this expression: 1 − n, with n = 2, 3, 4, 5,…..

Type of Consecutive Integers

There are mainly three types of consecutive integers:

• Normal consecutive integers (2, 3, 4, 5, ……)

• Even consecutive integers (2, 4, 6, 8, ……..)

• Odd consecutive integers (3, 5, 7, 9, ………)

Even Consecutive Integers

Consecutive even integers are the set of integers such that each integer in the set differs from the previous integer by a difference of 2 and each integer is divisible by 2.

Consecutive even integers are even integers that follow each other by a difference of 2. If x is an even integer, then x + 2, x + 4, x + 6 and x + 8 are consecutive even integers. 

Examples:

4, 6, 8, 10, …

-6, -4, -2, 0, …

124, 126, 128, 130, ..

You can represent consecutive even integers with the following expression: 2n + 2 with n = 0, 1, 2, 3….

Odd Consecutive Integers

Consecutive odd integers are the set of integers such that each integer in the set differs from the previous integer by a difference of 2 and each integer is an odd number.

Consecutive odd integers are odd integers that follow each other by the difference of 2. If x is an odd integer, then x + 2, x + 4, x + 6 and x + 8 are consecutive odd integers.

Examples:

1,3, 5, 7, 9, 11,…

-7, -5, -3, -1, 1,…

-25, -23, -21,….

You can represent consecutive odd integers with the following expression: 2n + 1 with n = 0, 1, 2, 3….

Consecutive Integers Formula

The consecutive integers formula is given by:

For odd consecutive integers

The general form of a consecutive odd integers formula is given by:

For even consecutive integers

The general form of a consecutive even integers formula is given by:

Where, “n” can be any integer.

Solved Examples: Consecutive Integers Problems

Example 1: John has a wooden board that is 5 feet long. He plans to make 4 shelves whose lengths are to be a series of consecutive even numbers. Find the length of each shelf in inches?

Solution:

Step 1: We know that consecutive even integers are even integers that follow each other by a difference of 2.

   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: Converting feet to inches

1 feet = 12 inches

Hence, 5 × 12 = 60 inches

Step 3: adding the 4 consecutive integers equal to 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 4: substitute the values of x

  length of first shelf = x = 12 inches

  length of second shelf = x + 2 = 14 inches

  length of third shelf = x + 4 = 16 inches

  length of fourth shelf =  x + 6 = 18 inches

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

Let’s check our answer 

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

Example 2: If the sum of three consecutive integers is 81, Find the three integers and  then find what is the product of the first and the third integer?

Solution:

Step 1: Let us assume the three consecutive integers: x, x + 1 and x + 2

Step 2 : Now, as given

x + x + 1 + x + 2 = 81

3x + 3 = 81

3x = 81 – 3

3x = 78

x = 78/3

x = 26

x + 1 = 27

x + 2 = 28

Product of the first and third integer = 26 × 28 = 728.

Quiz Time

Consecutive integers problems

The sum of two consecutive integers is 120. Find the value of the smaller integer.

The sum of two consecutive odd integers is 60. What are the integers?

Facts

The term consecutive numbers is used to frame word problems.

The sum of any two consecutive numbers is always odd. Example, 5 + 6 = 11