Sum of Odd numbers
2026-02-28 06:17 Diff
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Last updated on August 5, 2025

Odd numbers are the numbers that are not divisible by 2; for example, 1, 3, 5, 7, 9, 11,… The sum of the consecutive odd numbers is the sum of the odd numbers. In this topic, we will learn about the sum of odd numbers.

What is the Sum of Odd Numbers?

The sum of odd numbers is adding the consecutive odd numbers together. The sum of odd numbers formula is sn = n2. By using the formula, we can easily calculate the sum of odd numbers from 1 to infinity. For instance, the sum of the first five consecutive odd numbers is: 1 + 3 + 5 + 7 + 9 = 25.

Sum of Odd Numbers Formula

Odd numbers are the numbers that are not divisible by 2; the sum of n odd numbers can be calculated using the formula:

The sum of n odd numbers = n2, where n is the number of odd numbers

For example, let’s find the sum of the first 5 odd numbers

The sum of n odd numbers = n2

Here, n = 5

So, the sum of the first five odd numbers = 52 = 25

Sum of Odd Numbers Proof

The sum of the first n odd numbers formula is n2. To understand how we get this formula, let’ look at the pattern of odd numbers. The general form of odd numbers is 2n -1 where the common difference is 2. 

For the sequence of odd numbers: 1, 3, 5, 7,… (2n - 1)

d = 2

So, sn = 1 + 3 + 5 + 7 + …. + (2n - 1)

The sum of n terms of an arithmetic sequence is: Sn = n/2 (2a + (n - 1)d)

Substituting the value of a and d, here, a = 1, d = 2(3 - 1 = 2)

Sn = n/2 (2 × 1 + n - 1)2)

= n/2 (2 + 2n - 2)

= n/2 × 2n

= n2

  So, the sum of n odd numbers is n2.

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Sum of Odd Numbers NOT Starting From 1

Now we will find the sum of odd numbers not starting from 1. So let’s find the sum of odd numbers from 11 to 60.
 

The sum of the first n odd numbers is: Sn = n2

The sum of odd numbers not starting from 1, Sn = (n/2)(a + l), where a is the first term and l is the last term. 

Here, we will find the sum of n odd numbers from 11 to 60, so the sequence is 11, 13, 15, 17, 19,…, 59

To find the sum, first we will find n. Here, a = 11 and d = 2

an = a + (n - 1)d

59 = 11 + (n - 1)2

59 = 11 + (2n - 2)

59 = 2n + 9

2n = 59 - 9

2n = 50

n = 50/2 

= 25

Here, n = 25

So, sn = n/2 (a + l)

Where l is the last term 

= 25/2(11 + 59) 

= 25/2 + (70) 

= 25 × 35 

= 875

Sum of n Natural Numbers Formula

The natural numbers are the counting numbers starting from 1. The sum of n natural numbers is: Sn = n(n + 1)/2

Let’s find the sum of the first 10 natural numbers.

Here, n = 1 and d = 1

So, sn =   n(n + 1)/2

= 10(10+ 1)/2 = 10 × 11/2

=110/2 = 55

Sum of Even Numbers

Even numbers are the numbers that are evenly divisible by 2. The sum of the first n even numbers (2, 4, 6, 8,…, 2n)can be calculated using the formula: Sn = n(n + 1), where n is the number of terms. 

Let’s find the sum of the first 10 even numbers.

Here, n = 10

So, S10 = 10(10 + 1) 

=10 × 11 

= 110

Sum of Squares of n Natural Numbers

The sum of the squares of n natural numbers can be calculated using the formula: (n(n + 1)(2n + 1))/6. Where n is the number of terms. 

For example, let’s find the sum of the squares of 5 natural numbers

The first 5 natural numbers are: 1, 2, 3, 4, 5

The sum of squares of n natural numbers: S5 = (5 (5 + 1) ((2×5) + 1))/6

= (5 (6) (11))/6

= 55

Sum of GP Formulas

The GP can be represented as a, ar, ar2, …, arn - 1, where a is the first term, r is the common ratio. The sum of n terms of GP: Sn = a(1 - rn)/(1 - r), where r ≠ 1. 

Sum of infinite terms of GP: Sn = a/(1 - r), where |r| < 1.

Sum of n Terms of an AP

In AP, the difference between any two consecutive terms will be the same. The sum of n terms of an AP is: Sn = (n/2)(2a + (n - 1)d), where ‘a’ is the first term and ‘d’ is the common difference.    

Tips and Tricks for the Sum of Odd Numbers

By mastering the sum of odd numbers, students can make calculations faster and improve their mental math skills. Here are some tips and tricks to master the sum of odd numbers. 

  • Memorize the formula to make the calculation easier: Sn = n2.
  • Identify the pattern: The sum of a few odd numbers is: 1, 3, 5, 7, 9, …, and their sum form perfect squares:
    1 = 12 
    1 + 3 = 4 = 22
    1 + 3 + 5 = 9 = 32
  • You can also use the sum of arithmetic progression formula Sn = (n/2)(2a + (n - 1)d) to find the sum of odd numbers in a sequence. 

Common Mistakes and How to Avoid Them in Sum of Odd Numbers

Students make errors when finding the sum of odd numbers. Here are some mistakes and the ways to avoid them in the sum of odd numbers.

Problem 1

Find the sum of the first 5 odd numbers?

Okay, lets begin

The sum of the first 5 odd numbers is 25.

Explanation

The first five odd numbers are: 1, 3, 5, 7, 9.

The sum of odd numbers can be calculated using the formula: Sn = n2
= 52 = 25.

Well explained 👍

Problem 2

Find the sum of odd numbers between 10 and 30?

Okay, lets begin

The sum of odd numbers between 10 and 30 is 200.

Explanation

The odd numbers between 10 and 30 are 11, 13, 15, 17, 19, 21, 23, 25, 27, 29.

Here, a = 11

l = 29

d = 2

an = a + (n - 1)d

29 = 11 + (n - 1)2

29 = 11 + 2n - 2

29 = 2n + 9

2n = 20

n = 20/2 = 10

So, the sum = n/2 (a + l)

= (10/2)(11 + 29)

= 5 × 40 

= 200

Well explained 👍

Problem 3

Find the sum of the first 5 odd numbers, starting from 5?

Okay, lets begin

The sum of the first 5 odd numbers starting from 5 is 45.

Explanation

The odd numbers starting from 5: 5, 7, 9, 11, 13

The sum of n terms = (n/2)(2a + (n - 1)d)

Here, n = 5

a = 5

d = 2

So, Sn = (5/2)(2 × 5 + (5 - 1)2) 

= (5/2)(10 + 8)

= (5/2) × 18

= 45

Well explained 👍

Problem 4

Find the sum of odd numbers from 51 to 71?

Okay, lets begin

The sum of odd numbers from 51 to 71 is 671.

Explanation

Here,

a = 51

l = 71

Number of terms(n) =  an = a + (n - 1)d

71 = 51 + (n - 1)2

71 = 51 + 2n - 2

2n = 22

n = 11

Finding the sum using the formula: (n/2)(a + l)

= 11/2(51 + 71)

= 11/2(122)

= 671

Well explained 👍

Problem 5

Find the sum of odd numbers from 1 to 99?

Okay, lets begin

The sum of odd numbers from 1 to 99 is 2500.

Explanation

Here, a = 1

l = 99

The number of terms can be calculated using = an = a + (n - 1)d

99 = 1 + (n - 1)2

99 = 1 + (2n - 2)

2n = 99 - (-1)

2n = 100

n = 50

The sum of n odd numbers: Sn = n2

502 = 2500

Well explained 👍

FAQs on the Sum of Odd Numbers

1.What are odd numbers?

Odd numbers are the numbers that are not evenly divisible by 2, such as 1, 3, 5, 7, … 

2.What is the formula for the sum of the n odd numbers?

The sum of the n odd numbers is calculated using the formula: Sn = n2

3.Sum of the first 10 odd numbers?

The sum of the first 10 odd numbers is 100. As the number of terms is 10, so sn = n2 → 102 100.

4.Is 0 an odd number?

No, 0 is not an odd number. 

5.What are the applications of the sum of odd numbers?

The sum of odd numbers is used to recognize patterns in sequences, solve problems related to geometry, and for mental math.

Important Glossaries for the Sum of Odd Numbers

  • Odd numbers: Numbers that are not divisible by 2 are considered odd numbers. For example, 1, 3, 5, 7, .. 
  • Sum: The result of adding two or more numbers is the sum
  • Arithmetic progression: An arithmetic progression is a sequence of numbers where the difference between any two consecutive numbers is always the same.
  • Common difference: The difference between any two consecutive numbers in an AP is the common, and the difference is known as the common difference.
  • Even number: The numbers that are divisible by 2 are even numbers, for example, 2, 4, 6, 8.

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Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

Fun Fact

: She loves to read number jokes and games.