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# Consecutive Integers

When we come across with numbers and its different types, we learn about integers. These are important type of numbers which include natural numbers, zero, and negative of natural numbers. In other words, integers are the superset of whole numbers and negative of all natural numbers. The set of integers is denoted by $\mathbb{Z}$. We can represent the set of integers in the following way :
$\mathbb{Z}$ = {... -3, -2, -1, 0, 1, 2, 3,...}
There is an important concept called consecutive integers which are used in various word problems in mathematics. Consecutive integers are the integers that follow a sequence, that is each subsequent number begin 1 more than a previous number. The consecutive numbers are the important operations among the number systems. Consecutive integers follow one after another without interruption.
Consecutive integers are represented by n, n+1, n+2..., where 'n' is any integer.

### Types of consecutive integers:

• Even consecutive integers
• Odd consecutive integers

Even consecutive numbers are those which start from 2 and are obtained by adding 2 successively, such as - 2, 4, 6, 8, 10, ....

On the other hand, odd consecutive number start from 3 and eventually can be obtained by adding 2, such as - 3, 5, 7, 9, ....

 Related Calculators Consecutive Integer Calculator Adding Integer Calculating Integers Multiplying Integer

## Even Consecutive Integers

When the continuous integer begins with an even number and the difference between the two numbers in the sequence is two.

Example for even consecutive integers are 12, 14, and 16

## Odd Consecutive Integers

When the continuous integer begins with an odd number and the difference between the two numbers in the sequence is two.

Example for even consecutive integers are 13, 15, and 17

## Consecutive Integer Problems

Use below widget to find consecutive integers.

Here are some consecutive integers problems -

### Solved Examples

Question 1: Find the three consecutive integers when their sum is 33.
Solution:

Sum of the consecutive integers are 33.

Let the consecutive integers are (v), (v + 1), (v + 2)

By given data

(v) + (v + 1) + (v + 2) = 33

3 v + 3 = 33

3 v = 33 - 3

3 v = 30

v = $\frac{30}{3}$

v = 10

The algebra consecutive integers are

v =10

v + 1= 11

v + 2 = 12

The consecutive integers are 10, 11, and 12.

Question 2: Find the three even consecutive integers when their sum is 66.
Solution:

Sum of the even consecutive integers are 66.

Let the even consecutive integers are (v), (v+2), (v+4)

By given data

(v) + (v + 2) + (v + 4) = 66

3 v + 6 = 66

3 v = 66 - 6

3 v = 60

v = $\frac{60}{3}$

v = 20.

The algebra consecutive integers are

v =20

v + 2= 22

v + 4= 24

The consecutive integers are 20, 22, and 24.

Question 3: Find the two odd consecutive integers whose sum is 48
Solution:

Sum of two odd consecutive integers are 48

Let the odd consecutive integers are (x), (x+2)

By given data:

(x) + (x +2) = 48

2 x + 2 = 48

2 x = 48 -2

2 x = 46

x = 23

The numbers are 23 and 25

The two odd consecutive integers are 23 and 25.

## What are Consecutive Integers?

The word consecutive means continuous. Consecutive integers mean continuous integers where they have difference 1 with their neighbor integer.

A consecutive integer defines the continuous integers with difference of 1. That is, consecutive integers are x, x + 1, x + 2, x + 3, ... Every integer has a difference of 1 with their neighbor.

Examples of consecutive integers:

2, 3, 4, 5...

-5, -4, -3, -2, ...

## Three Consecutive Integers

Three consecutive numbers are one of the part in math. Numbers which follow each other in order, without spaces, from smallest to largest is known as consecutive numbers. For example 12, 13 and 14 are three consecutive numbers.

Below you could see examples for three consecutive integers and sum of three consecutive integers.

### Solved Examples

Question 1: The sum of the least and greatest of 3 consecutive integers is 160. What are the values of the three numbers?
Solution:
Step 1: Let us assume the integers are x, x + 1 and x + 2 .

Step 2: And we know that , the least and greatest of the 3 consecutive integer is 160 .

Step 3: So, we can arrange the equation, that is x + (x + 2) = 160 .

Step 4: Here we need to solve this equation. So 2x + 2 = 160. (Subtract 2 on both the sides)

Step 5: Therefore x = 79. Therefore the least integer is 79.

Step 6: Substitute the x value is x + 1, x + 2 .

Step 7: Therefore, 79 + 1 = 80 and 79 + 2 = 81.

Step 8: Therefore , the values of 3 integers are 79, 80 and 81.

Question 2: The sum of the least and greatest of 3 consecutive integers is 180. What are the values of the three numbers?
Solution:
Step 1: Let us assume the integers are x, x + 1 and x + 2 .

Step 2: And we know that , the least and greatest of the 3 consecutive integer is 180 .

Step 3: So, we can arrange the equation, that is x + (x+2) = 180 .

Step 4: Here we need to solve this equation. So 2x + 2 = 180. (Subtract 2 on both the sides)

Step 5: Therefore x = 89. Therefore the least integer is 89.

Step 6: Substitute the x value is x+1, x+2 .

Step 7: Therefore, 89+1 = 90 and 89+2 = 91.

Step 8: Therefore , the values of 3 integers are 89, 90 and 91.

Question 3: Sum of three consecutive integers is 18. Find the integers?
Solution:
Consecutive integers = x+x+1+x+2 = 18

3x+3=18

Subtract both sides by 3 we get,

3x+3-3 = 18-3

3x = 15

Divide both sides by 3 we get,

$\frac{3x}{3}$ = $\frac{15}{3}$

X = 5

Therefore, the unknown number is 5.

Then, three consecutive integers are,

X = 5

X+1 = 6

X+2 = 7

Three Integers are 5, 6 and 7

## Two Consecutive Integers

The two consecutive numbers are the numbers in continuous order, without gaps, from smallest to largest. For example 2 and 3 are two consecutive numbers.

Below you could see examples for two consecutive integers and sum of two consecutive integers

### Solved Examples

Question 1: The sum of two consecutive integers is 15 . Find the two numbers.
Solution:
Let x be one of the consecutive numbers.

x + 1 is the other consecutive number

The sum of these two consecutive numbers is x+(x+1)= x+x+1= 2x+1

Given the sum is 15

Therefore 2x + 1 = 15

=> 2x = 15-1 = 14

=> 2x = 14

=> x = $\frac{14}{2}$ =7

=> x = 7

The consecutive number is x + 1 = 8

The two consecutive numbers are 7, 8.

Question 2: Addition of two consecutive odd integers is 104. What is the value of two odd integers?
Solution:
Step 1: Write the given two consecutive even integer = 104.

Step 2: Let us assume the odd integer as ‘x’.

Step 3: Then the formula is given by ‘x’ is ‘x’ + 2.

Step 4: Simplifying the given expression, we get,

x + (x + 2) = 104

x + x + 2 = 104

2x + 2 = 104

2x = 104 – 2

2x = 102

Divide by 2 on both the sides

$\frac{2x}{2}$ = $\frac{102}{2}$

x = 51

The next consecutive odd integer for 51 is 51 + 2 = 53.

Therefore, the two consecutive odd integers for 167 is given by 51 and 53.

## Consecutive Integer Word Problems

Below you could see consecutive integer word problems

### Solved Examples

Question 1: The Sum of the two consecutive integers is 75. What are the values of the 2 integers?
Solution:
Let the first integer = n

Second integer = n+1

Sum of the two consecutive integers = 75

n+(n+1) = 75

2n+1 = 75

2n = 75-1

2n = 74

n = $\frac{74}{2}$

n = 37

n+1 = 37+1= 38

Hence, the two integers are 37, 38

Question 2: Find four consecutive integers such that, the sum of the two larger integers is 65.
Solution:

Let, 1st integer  = n

2nd = n + 1
3rd = n + 2
4th = n + 3

Large two integers are n + 2,  n + 3

According to question:

(n + 2) + (n + 3) = 65

2n + 5 = 65

2n = 65 - 5

2n = 60

n = $\frac{60}{2}$

n = 30 (first integer)

Second integer = n + 1 = 30 + 1 = 31

Third integer = n + 2 = 30 + 2 = 32

and  forth integer = n + 3 = 30 + 3 = 33

Hence four consecutive integers are, 30, 31, 32, 33.

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