Consecutive integers are those numbers that follow each other. They follow in a sequence or in order. For example, a set of natural numbers are consecutive integers.

Consecutive means an unbroken sequence or following continuously so that consecutive integers follow a sequence where each subsequent number is one more than the previous number. In a set of consecutive integers (or in numbers), the mean and median are equal.

If x is an integer, then x + 1 and x + 2 are two consecutive integers.

**Examples: **

4, 5, 6, 7, 8,..

-1, 0, 1, 2, 3, 4, 5, 6,..

-20, -19, -18, -17,…

## Consecutive Integers in Maths

As discussed in the introduction, in Maths, the numbers that follow each other in an order are called consecutive numbers or consecutive integers. These integers go from smallest to the highest, i.e. in ascending order. Some of the examples are:

- -4,-3,-2, −1, 0, 1, 2, 3, 4
- 10,11,12,13,14,15,16
- 100,101,102,103,104,105,106

From the above examples, we can see, the integers follow each other in a sequence. The difference between preceding and succeeding integers is always equal to 1.

- 4-3 = 1
- -2-(-3) = 1
- 101-100 = 1

### Odd Consecutive Integers

Consecutive odd integers are odd integers that follow each other and they differ by 2. If x is an odd integer, then x + 2, x + 4 and x + 6 are consecutive odd integers.

**Examples:**

5, 7, 9, 11,…

-7, -5, -3, -1, 1,…

-25, -23, -21,….

### Even Consecutive Integers

Consecutive even integers are even integers that follow each other and they differ by 2. If x is an even integer, then x + 2, x + 4 and x + 6 are consecutive even integers. Consecutive even integers differ by two.

**Examples:**

4, 6, 8, 10, …

-6, -4, -2, 0, …

124, 126, 128, 130, ..

## Consecutive Integers Formula

The given formulas are the algebraic representations of consecutive integers.

The formula to get a consecutive integer is **n + 1, **

**For odd consecutive integers:**

The general form of a consecutive odd integer is **2n+1**,

**For even consecutive integers:**

The general form of a consecutive even integer is** 2n**,

Where

“n” can be any integer.

## Product of Three Consecutive Integers

If there are three consecutive integers, say a, b and c, then their product is given by:

a x b x c = abc

Example:

- 1 x 2 x 3 = 6
- 7 x 8 x 9 = 504
- 9 x 10 x 11 = 990

From the above three examples, we can conclude an interesting fact that the product of any three consecutive integers, is always divisible by 6.

**How to Find integers? **

Let us say, x, x+1 and x+2 are three consecutive integers, then the product of these three consecutive integers are given by:

x (x+1) (x+2) = x (x2 + x + 2x + 2)

= x3 + 3x2 +2x

By putting the above equation equal to the product of three consecutive integers and solving for x, we can determine the value of required integers.

## Properties

The following are the properties of consecutive numbers:

- The difference between any two consecutive odd or even integers is 2.
- There will be accurately one number divisible by n in any set of n consecutive integers. For example, any four integers in a row must have a multiple of 4; any 19 integers will have one multiple of 19 and so on. Consider a set of three consecutive integers: {–1, 0, +1}, here, multiple of 3 does not exist. This is the special case when it turns out to zero. We know that when an integer is multiplied with zero, it gives 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 is possible to have 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 even number of consecutive integers, the even and odd integers will be in equal number.
- If m is an odd number, then the total sum of m consecutive integers will be divisible by m. For example, for any five integers in a row, the sum is divisible by 5., etc.,

## Solved Problems

In Maths, there are many numerical and word problems that can be solved using this concept. Following are the example problems based on the concept of consecutive integers.

**Example 1: Suppose the sum of four consecutive odd integers is 184. Find the smallest number.**

Solution:

Let x, x+2, x+4 and x+6 be the four consecutive odd integers.

According to the given,

x + x + 2 + x + 4 + x + 6 = 184

4x + 12 = 184

4x = 184 – 12

4x = 172

x = 172/4

x = 43

Hence, the smallest number is 43.

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

Solution:

Consider three consecutive integers: x, x + 1 and x + 2

According to the given statement,

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 last integer = 26 × 28 = 728.

**Example 3: Find three consecutive integers that add up to 51.**

Solution: Suppose the three consecutive numbers are x,x+1,x+2

Given, sum of the numbers is equal to 51.

∴x+x+1+x+2=51

3x+3=51

3x=48

x=16

Therefore,

x=16,

x+1=16+1=17,

x+2=16+2=18

Thus, the numbers are 16,17,18.

**Example 4: The sum of five consecutive integers is 100. find the third number.**

Solution: Let the five consecutive integers be x,x+1,x+2, x+3 and x+4

As per the given questions,

X+x+1+x+2+x+3+x+4 = 100

5x + 10 = 100

5x = 90

X = 18

Therefore, the third integer is x+2 = 18+2 = 20

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