Sum of odd numbers – Explanation and Examples

What is an odd natural number?

A system of natural numbers that is not a multiple of 2 is known as a set of odd natural numbers. For example, 1, 3, 5, 7, …. are odd natural numbers because upon division by 2, they give a fractional form. Odd numbers are placed alternatively in a series of natural numbers, and by using an arithmetic progression, we can calculate the sum of odd natural numbers very quickly.

What is the formula to find the sum of odd numbers?

Now that you recognize odd numbers, we will derive the formula for the sum of first n odd numbers by arithmetic progressions. Arithmetic Progression (A.P) is a sequence of numbers in which the difference between any two consecutive numbers remains constant. So, for example, the series of odd natural numbers, i.e., 1, 3, 5, 7, 9, …. is an A.P because the difference between two consecutive odd numbers remains 2.

Here is how AP can help us derive the answer.

Let the sum of first n odd numbers be S_n

S_n = 1 + 3 + 5 + 7 + 9 + ………………….. + (2n-1)                             – equation 1

Wondering how we found the last term of this AP series?

According to the above series, a = 1 and d = 2

The formula for finding the n^th term of an AP is a + (n – 1)d

Therefore, upon putting these values in the n^th formula, we get the last term as (2n – 1)

By arithmetic progression, we know the formula to find the sum of n numbers;

S_n = ½ x n [2a + (n-1) d]                                                                – equation 2

In this equation,

n = total number of digits in the series

a = first digit of the A.P

d = common difference in the A.P

With respect to equation 1, we know that;

a = 1, d= 2

Upon substituting these values in equation 2 with respect to equation 1;

S_n = (n/2) x (a + l)

S_n = (n/2) x (1 + 2n – 1)

S_n = (n/2) x (2n)

S_n = n²

Therefore, sum of first n odd numbers (S_n) = n²

Solved examples

Question 1: How many odd numbers are there between 1 to 100? Find their sum.

Solution:

We know that there is a total of 50 odd numbers between 1 and 100. According to the formula for the sum of first n odd numbers S_n = n²

Here, n = 50

S₅₀ = (50)²

S₅₀ = 2500

Hence, the sum of odd numbers between 1 and 100 is 2500.

Question 2: Prove that the formula and sum of first n natural odd numbers manually give the same answer.

Solution:

Starting from 1, suppose we have n odd natural numbers, 1, 3, 5, 7,…… (2n-1)

So,

The sum of the first odd natural number is 1

Sum of first two odd natural numbers = 1 + 3 = 4 = 2²

Sum of first three odd natural numbers = 1 + 3 + 5 = 9 = 3²

Sum of first four odd natural numbers = 1 + 3 + 5 + 7 = 16 = 4²

Hence, the formula and sum of first n natural odd numbers manually give the same answer.

Question 3: Find the odd numbers between 12 and 24 and find their sum.

Solution:

The series of odd numbers between 12 and 24 is 13, 15, 17, 19, 21, 23

here, a = 13, n = 6, d = 2

Putting these values in S_n = ½ x n [2a + (n-1) d]

S₆ = ½ x 6 [2 x 13 + (6 -1) 2]

S₆ = ½ x 6 [26 + 10]

S₆ = 3 x 36

S₆ = 108

So, sum of the series 13, 15, 17, 19, 21, 23 is 108.