Consecutive Integers

Consecutive Integers are the integers that follow each other, i.e. while continuously writing integers they come next to each other. they have a difference of one(1). For example, ...-3, -2, -1, 0, 1, 2, 3,... this is a sequence of consecutive integers. Apart from that natural numbers are also called consecutive integers because they are all integers and the difference between two consecutive natural numbers is always 1.

Thus, we can say that consecutive integers are the number that follows a regular pattern of writing and there is a fixed difference between any two consecutive integers, i.e. they have a difference of One(1). We represent any two consecutive integers as n, n + 1 where n ϵ Z.

In this article, we will learn about Consecutive Integers, Consecutive Even Integers, Consecutive Odd Integers, Examples, and others in detail.

Before learning in detail about Consecutive Integers let's first understand what is the meaning of Consecutive in maths.

Consecutive Meaning

Consecutive in Mathematics is defined as the number that comes next to each other. Suppose we have to write consecutive natural numbers then they are 1, 2, 3,..., suppose we have to write consecutive whole number then they are, 0, 1, 2, 3,... Now one thing to note is that their are no consecutive rational numbers or consecutive irrational numbers, because between any two rational numbers and irrational numbers there are infinite rational and irrational number.

What are Consecutive Integers?

Consecutive integers are the integers in mathematics that follow each other. They have a difference of 1 and they follow a regular pattern. The consecutive integers are always arranged in ascending order and written from smallest to largest.

Sequence of consecutive integer is represented as, n, n + 1, n + 2, n + 3, ...

General Term of Consecutive Integer = n where n ϵ Z

Examples of Consecutive Integers

Some examples of consecutive integers include,

• 0, 1, 2, 3, 4, 5, ...
• -5, -4, -3, -2, -1, 0, ...
• -2, -1, 0, 1, 2,...
• 100, 101, 102, 103, ...

From any of the example we can see that the difference between two consecutive integers is always one.

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

Consecutive Even Integers

We know that even integers are multiples of 2. So, if we list the set of even numbers in ascending order, they may be expressed as ...,-4, -2, 0, 2, 4, 6, 8, 10, ...

Thus, they are even consecutive number. We can see that the difference between each consecutive even integer is 2. The even consecutive number are represented in general form as, 2n, 2n + 2, 2n + 4, ... where n ϵ Z

For example,

• ..., 0, 2, 4, 6, ...
• ..., -4, -2, 0, 2, 4, ....

Consecutive Odd Integers

We know that odd integers that on dividing by 2 gives one(1) as remainder. So, if we list the set of odd numbers in ascending order, they may be expressed as ...,-3, -1, 0, 1, ...

Thus, they are odd consecutive number. We can see that the difference between each consecutive odd integer is 2. The odd consecutive number are represented in general form as, 2n + 1, 2n + 3, 2n + 5, ... where n ϵ Z

For example,

• ..., 1, 3, 5, 7, ...
• ..., -3, -1, 0, 1, 3, ....

Consecutive Integers Formula

The formula to calculate the consecutive integers is very basic and uses the concept of basic algebraic expressions.

For any given integer n the formula for the consecutive integer is n + 1

For Even Consecutive Integers

• The formula to calculate the even consecutive odd integer is 2n

For Odd Consecutive Integers

• The formula to calculate the odd consecutive odd integer is 2n +1

Consecutive Positive Integers

The consecutive positive integers are the consecutive numbers that are positive in nature, i,e, they are the positive natural numbers, for example 1, 2, 3, 4, 5,... are consecutive positive numbers. The difference between two consecutive integers is constant and is always equal to 1.

Three Consecutive Integers

Product of three consecutive integers (apart from 0) is always divisible by 6 this is shown as, suppose we have three consecutive integers then they are represented as,

n, n + 1, n + 2

Now the product of these three integers is,

(n)(n +1)(n + 2) = n³ + 3n² + 2n

The above expression is always divisible by 6 and can be proved using Mathematical Induction. Thus, we can say that product of three consecutive integers is always divisible by 6.

For example,

• 1 × 2 × 3 = 6 {Divisible by 6}
• 5 × 6 × 7 = 420 {Divisible by 6}
• 11 × 12 × 13 = 1716 {Divisible by 6}, etc

Properties of Consecutive Integers

Various properties of the consecutive integers are,

• Difference between two consecutive integers is always constant and is equal to 1.
• Difference between two consecutive even integers is always constant and is equal to 2.
• Difference between two consecutive odd integers is always constant and is equal to 2.
• Product of three consecutive integers (apart from 0) is always divisible by 6.

Read More,

• Irrational Numbers
• Rational Numbers
• What is Arithmetic?

Consecutive Integers Solved Examples

Example 1: Find three consecutive integers after 64.

Solution:

Three consecutive integers sequence after x will be x + 1, x + 2, and x + 3

So, three consecutive integers after 64 are,

= 64 + 1, 64 + 2, and 64 + 3

= 65, 66, and 67 

Therefore, three consecutive integer after 64 will be 65, 66, and 67 

Example 2: Find 5 consecutive even integers of -10?

Solution:

Consecutive Integer after - 10,

Therefore, Let - 10 be x

So the next five consecutive even integers will be

= x + 2, x + 4, x + 6, x + 8, and x + 10

=  (-10 + 2), (-10 + 4), (-10  + 6), (-10 +8), and ( -10 +10) 

=  -8, -6, -4, -2, and 0

Therefore, the five even consecutive integers will be have -10, -8, -6, -4, -2, 0.

Example 3: Find the smallest number if the sum of four consecutive odd integers is 288? 

Solution:

Sum of four consecutive odd integers are 288

So, four consecutive odd integers sequence will be  x, x + 2, x + 4 and x + 6 

According to question,

x + x + 2 + x + 4 + x + 6 = 288

4x + 12 = 288

4x = 288 – 12

4x = 276

x = 276/4

x = 69

Hence, the smallest number will be 69.

Example 4: If the sum of four consecutive integers is 302, then what is the product of the first and the third integer?

Solution:

Lets assume four consecutive integers are x, x + 1, x + 2, x + 3 

Sum of four consecutive integers are 302

Therefore,

x + x + 1 + x + 2 + x + 3 = 302

4x + 6 = 302

4x = 302 – 6

4x = 296

x = 296/4

x = 74

So,

• First integer is, x = 74
• Third integer is, x + 2 = 76

Product of the first and third integer = 74 × 76 = 5624

Example 5: Find two consecutive positive integers sum of whose squares is 265?

Solution:

Two consecutive positive integers sum of whose squares is 265

Let's assume one integer be x, then the other integer is x + 1, 

So,

x² + (x + 1)² = 265

x² + x² + 1 + 2x  = 265 [Algebraic Identity (a + b)² = a² + 2ab + b²]

2x² + 2x + 1 = 265

2x² + 2x + 1 - 265 = 0

2x² + 2x - 264 = 0

x² + x - 132 = 0            

x² + 12x - 11x - 132 = 0

x(x + 12) - 11 (x + 12) = 0

(x +12) (x -11) = 0

x = -12 or x = 11

Since, we need a positive integer, 

Therefore x = -12 is rejected.

Thus, x = 11

x + 1 =  11 + 1 = 12 

Hence first two positive consecutive integers are 11, 12.

Practice Questions on Consecutive Integers

Q1: Find the next 10 consecutive integers from -20

Q2: Find the sum of first 5 consecutive even numbers

Q3: Find the sum of first 7 consecutive even numbers

Q4: Find the product of three consecutive integers 9, 10 and 11