Consecutive Numbers

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What is the Meaning of Consecutive Numbers?

Before understanding consecutive numbers let us look into consecutive meaning in math.

Consecutive means following each other continuously in a series or a sequence. In Mathematics numbers that follow each other in a series are termed consecutive numbers.

Consecutive numbers meaning is “The numbers which continuously follow each other in the order from smallest to largest.” 

Consecutive numbers example are 

Consecutive numbers from 1 to 8 are 1, 2, 3, 4, 5, 6, 7, 8. Here the difference between each number is 1.

Consecutive numbers from 80 to 90 with difference as 2 are 80, 82, 84, 86, 88, 90. Here the difference between each number is 2. This type of consecutive number system where each number is an even number and the difference between each number as 2 is called consecutive even integers which we will discuss in detail in the latter part of this chapter.

Consecutive numbers from -1 to 7 are -1, 0, 1, 2, 3, 4, 5, 6, 7. Here the difference between each number is 1.


What Does Consecutive Natural Numbers Mean?

Natural numbers are the numbers that are used for counting and ordering. Natural numbers include all whole numbers except 0.

In the mathematical form the natural numbers are written as follows:

N = {1, 2, 3, 4, 5, 6, 7, 8, 9, ………….}

So consecutive natural numbers are the numbers that continuously follow each other in order from the smallest number to the largest number.

Ex: The consecutive natural numbers from 10 to 15 are 10, 11, 12, 13, 14, 15.

The consecutive numbers from -5 to -1 is not a consecutive natural number because these numbers don't fall under natural numbers. These numbers are integers.


What is Meant by Consecutive Numbers with Even Integers?

An even number is one that is divisible by two and leaves a 0 as a remainder.

So Consecutive Numbers with Even integers or simply called consecutive even integers is a sequence where the numbers continuously follow each other in the order from the smallest number to the largest number with the difference between each number is 2 and each number is divisible by 2. 


Consecutive Integers Examples

2, 4, 6, 8, 10, 12 are consecutive even integers between the numbers 2 to 12.

-8, -6, -4, -2 is consecutive even integers between the numbers -8 to -2.


What is Meant by Consecutive Numbers with Odd Integers?

A number that is not divisible by two is called an odd number. In the case of an odd number, the remainder is always 1.

So consecutive odd integers is a sequence where the numbers continuously follow each other in the order from the smallest number to the largest number with the difference between each number is 2 and each number not divisible by 2.


Consecutive Integers Examples

The consecutive odd integers between 1 to 15 are 1, 3, 5, 7, 9, 11, 13, 15.

The consecutive odd integers between -11 to -1 are -11, -9, -7, -5, -3, -1.


Properties of Consecutive Numbers

  • Any pair of predecessors and successors have a fixed difference. If we denote the first number as n, then n, n+1, n+2, n+3, n+4 and so on will be the consecutive numbers in the sequence.

Ex: The sequence 2, 8, 14, 20 is Consecutive Numbers with a fixed difference of 6. Similarly, 2, 7, 13, 20 are not Consecutive Numbers because the fixed difference is changing from number to number.

  • The difference between any two consecutive odd numbers is 2.

Ex: In the sequence of consecutive odd integers 3, 5, 7, 9, 11 the difference between any two consecutive odd numbers is 2 i.e. 5 - 3 = 2, 7 - 5 = 2 and so on.

  • The difference between any two consecutive even numbers is 2.

Ex: In the sequence of consecutive even integers 12, 14, 16, 18, 20 the difference between any two consecutive odd numbers is 2 i.e. 14 - 12 = 2, 16 - 14 = 2 and so on.

  • In a row, any n numbers will have a multiple of n.

Ex: Consider consecutive numbers 44, 45, 46, 47, 48, 49, 50, 51, 52 which are having 9 numbers in the sequence. So according to the property, anyone number should be a multiple of 9, where 45 is a multiple of 9. 45 = 9*5.

  • The sum of n consecutive numbers is divisible by n if n is an odd number.

Ex: Let us consider a consecutive odd number sequence 5, 7, 9, 11, 13, 15, 17 which are having 7 numbers in the sequence. So according to the property, the sum of this consecutive odd number sequence should be divisible by 7. 

5+7+9+11+13+15+17 = 77. 

77/7 = 11 which satisfies this property. 

Here we have to note that this property applies only when n is an odd number i.e only if the sequence is made of odd numbers.

  • The sum of two consecutive numbers is always an odd number.

Ex: Consider a consecutive number sequence 1, 2, 3, 4, 5, 6, 7, 8, 9. Here the sum of the two consecutive numbers will always be an odd number. 1+2 = 3, 2+3 = 5, 3+4 = 7, and so on.

  • The sum of two consecutive odd numbers is always an even number.

Ex: Consider a consecutive odd number sequence 5, 7, 9, 11, 13, 15, 17. Here the sum of two consecutive odd numbers will always be an even number. 5+7 = 12, 7+9 = 16, 9+11 = 20, and so on.

  • The sum of two consecutive even numbers is always an even number.

Ex: Consider a consecutive even number sequence 12, 14, 16, 18, 20, 22, 24. Here the sum of two consecutive even numbers will always be an even number. 12+14 = 26, 14+16 = 30, 16+18 = 34, and so on.


Problems with Consecutive Numbers

1) What is a consecutive number in math for natural numbers between 1 to 25?

Ans: The consecutive number for the natural numbers between 1 to 25 is as follows:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25.

2) Write the consecutive even integers between 20 to 40. 

Ans: The consecutive even integers between 20 to 40 is as follows:

20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40.

3) Find the sum of two consecutive numbers 8 and 9.

Ans: The sum of two consecutive numbers is 8+9 = 17.


Conclusion 

  • Just remember that the sequence of a consecutive number series depends on the first number which we are representing. Say if we are representing the first number as n then the consecutive numbers will be n+1, n+2, n+3, n+4, and so on.

  • The general form to represent consecutive even numbers is 2n where n is an integer. 

  • The general form to represent consecutive odd numbers is 2n+1 where n is an integer.

FAQs (Frequently Asked Questions)

1. What is a Consecutive Number in Math?

Ans: Numbers that continuously follow each other in the order from smallest to largest are called consecutive numbers.

2. What are Consecutive Even Integers?

Ans: Consecutive even integers is a sequence where the numbers continuously follow each other in the order from the smallest number to the largest number with the difference between each number is 2 and each number is divisible by 2.

3. What is the Sum of Two Consecutive Numbers?

Ans: The sum of two consecutive numbers will always be an odd number.

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