 # Consecutive Integers

## Introduction

When you start counting natural numbers what are you doing 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 any integer.

For example: 23, 24, 25

Look at the following two sets. 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 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

 n + 1

For odd consecutive integers:

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

 2n+1

For even consecutive integers:

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

 2n

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 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, 14inch, 16inch and 18inch.

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

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

2. 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

1. What are the Consecutive Integers Properties

Answer : The following are the consecutive integers properties numbers:

• The difference between any two consecutive either odd or even integers is 2.

• Any set of integers has accurately one number divisible by n. For example, any six integers in a row must have a multiple of 6; any 15 integers will have one multiple of 15 and so on.

•  Consider a set of three consecutive integers: {–1, 0, +1}, here we observe multiple of 3 does not exist. This is the special case when it turns out to zero.

• Depending upon the set which has been started, there might be two even numbers and one odd number, or two odd numbers and one even number in a set of 3 consecutive integers. In a set of 4 consecutive integers,it has two even and two odd numbers. Depending on the starting value, if a set has an odd number of consecutive integers, there will be a chance of more evens or more odds. But if a set has an even number of consecutive integers, the even and odd integers will be in equal number.

• If x is an odd number, then the total sum of x consecutive integers will be divisible by x. For example, for any three integers in a row, the sum is divisible by 3, etc.,

2. How consecutive integers are represented in algebraic representations

Answer : It is easy to recognize a set of consecutive integers in plain numbers.

But the algebraic representation of consecutive integers is different. The following are examples of representations of consecutive integers in algebraic form:

{n, n + 1, n + 2, n + 3, n + 4, n + 5, n+ 6}

{n - 4, n - 3, n - 2, n - 1, n, n + 1, n + 2, n + 3, n + 4}

{n + 11, n + 12, n + 13, n + 14, n + 15, n + 16, n+ 17}

For simplicity, let’s pick n = 5.  The first set becomes the set of integers from 5 to 11 ; the second, from 1 to 8; and the third, from 16 to 22.

6. How Many Digits at the Maximum Count can a Set of Consecutive Integers have?

There is no limitation in the maximum count of digits in a set of consecutive integers. 1, 3, 5, 7... and 234, 456, 567, 789… both these sets are considered to be consecutive as they don’t have any gaps breaking the number sequence.