Sum of Odd Numbers - Formula, Proof | Sum of First n Odd Numbers (2024)

Odd numbers are defined as any number that is not a multiple of 2 or has its units place digit to be one of the digits 1, 3, 5, 7, or 9. The sum of odd numbers is the total summation of the odd numbers taken together for any specific range given.

The sum of odd numbers can be calculated easily using the sum of the arithmetic progression formula. We will be learning about the sum of odd numbers and the sum of first n odd numbers using formulas and examples in this article.

1.	What is Sum of Odd Numbers?
2.	Sum of n Odd Numbers Formula
3.	Sum of First n Odd Numbers Proof
4.	Sum of Odd Numbers NOT Starting From 1
5.	FAQs on Sum of Odd Numbers

What is Sum of Odd Numbers?

The sum of odd numbers is defined as the summation of odd numbers taken together and added up to calculate the result. Let us examine the sum of odd numbers starting from 1 to a certain number to derive the sum of odd numbers formula.

The sum of odd numbers from 1 to 5 = 1 + 3 + 5 (3 numbers) = 9 = 3²
The sum of odd numbers from 1 to 7 = 1 + 3 + 5 + 7 (4 numbers) = 16 = 4²
The sum of odd numbers from 1 to 9 = 1 + 3 + 5 + 7 + 9 (5 numbers) = 25 = 5²
The sum of odd numbers from 1 to 11 = 1 + 3 + 5 + 7 + 9 + 11 (6 numbers) = 36 = 6²

By the above observation, we can derive a formula that "the sum of first n odd numbers is n²".

We can derive this formula involving the concept of arithmetic progression discussed in the upcoming sections.

Sum of n Odd Numbers Formula

We know that the general form of an odd number is (2n - 1) where n ≥ 1 is an integer. Also, consecutive odd numbers have a common difference of 2. Therefore, the series of odd numbers form an arithmetic progression. The sum of n odd numbers formula is described as follows,
Sum of n odd numbers = n² (we will prove this in the next section) where n is a natural number and represents the number of terms.

Thus, to calculate the sum of the first n odd numbers together without actually adding them individually, we can use the sum of n odd numbers formula i.e., 1 + 3 + 5 +...........n terms = n². Let's derive this formula now.

Sum of First n Odd Numbers Proof

Let us now derive the sum of n odd natural numbers formula. We know that the sequence of odd numbers is given as 1, 3, 5, ... (2n - 1) which forms an arithmetic progression with a common difference of 2. Let the sum of the first n odd numbers be represented as S_n = 1 + 3 + 5 +...+ (2n - 1). Here 1 represents the first odd number and (2n - 1) represents the last odd number.

Thus, the first term (a) = 1, the last term (l) = 2n - 1 and the common difference (d) = 2.

The sum of n terms of an AP is given by the formula S_n= n/2 × [a + l].

By substituting the values of 'a' and 'l' in the above formula we get,

S_n = n/2 × [1 + (2n - 1)]
S_n= n/2 × [2n]
S_n= n × n
S_n= n²

Thus, when n = 1, S₁ = 1² = 1
when n = 2, S₂ = 2² = 4
when n = 3, S₃ = 3² = 9

Therefore, we have proved that the sum of first n odd numbers is equal to n². Let's take an example to understand this.

Example: Find the sum of odd numbers 1 to 50.

We know that there are 25 odd numbers between 1 to 50. Thus, by using the sum of n odd numbers formula which is n², we get, S₂₅ = 25² = 625.

Sum of Odd Numbers NOT Starting From 1

If we have to find the sum of odd numbers where the starting digit is NOT 1, say we have to find the sum of odd numbers from 7 to 50. If we still want to use n² formula, we have to do it this way:

Required sum = (the sum of odd numbers from 1 to 50) - (the sum of odd numbers from 1 to 5)
= 25² - 3² (∵ there are 25 odd numbers from 1 to 50 & there are 3 odd numbers from 1 to 5).
= 625 - 9
=616

Alternatively, it is better to use the concept of the arithmetic progression instead of n² formula. For this, we list out the odd numbers from 7 to 50 which are 7, 9, ..., 49. Here 49 is the nth term of AP whose first term is a = 7 and the common difference is d = 2. Now, we will first find 'n'.

a_n = a + (n - 1) d
49 = 7 + (n - 1) 2
49 = 7 + 2n - 2
2n = 44
n = 22

Now, the sum of 22 terms of AP is:

S_n = n/2 (a + l)
= 22/2 (7 + 49)
=616

Note that we got the same answer using both methods.

☛Related Topics:

Even or Odd Calculator
Even and Odd Numbers

FAQs on Sum of Odd Numbers

What is the Meaning of Sum of Odd Numbers?

The sum of odd numbers is defined as the addition or summation of all the odd numbers present in a given range. For example, to calculate the sum of odd numbers between 1 to 10 we will consider all the odd numbers in this range and add them. 1 + 3 + 5 + 7 + 9 = 25.

What is the Sum of Odd Numbers Formula?

We know that the series of odd numbers are always in AP as the common difference between them is 2.

The sum of odd numbers formula is S_n= n/2 × [a + l] where 'a' is the first odd number, 'l' is the last odd number and 'n' is the number of odd numbers present in that range.
Another formula to calculate the sum of first n odd numbers is S_n= n². But to use this formula, the starting odd number should always be 1.

How to find Sum of Odd Numbers?

The sum of odd numbers can be calculated using the formula S_n= n/2 × [a + l] where 'a' is the first odd number, 'l' is the last odd number and 'n' is the number of odd numbers or S_n= n². To calculate the sum of odd numbers between 1 to 20 we will use S_n= n² where n = 10 as there are 10 odd numbers between 1 to 20. Thus, S₁₀ = 10² = 100.

What is the Formula for the Sum of the First n Odd Numbers?

The formula for the sum of the first n odd numbers is given as S_n= n² where n represents the number of odd numbers. i.e., n² is the sum of odd numbers from 1 to (2n - 1).

How to Find Sum of First n Odd Numbers?

To find the sum of first n odd numbers we can use the formula S_n= n². For example, to calculate the sum of odd numbers between 1 to 10, we know that n = 5. Thus, S₅ = 5² = 25.

What is the Sum of Odd Numbers From 1 to 100?

Note that there are 50 odd numbers from 1 to 100. i.e., n = 50. Therefore, the sum of odd numbers from 1 to 100 = n² = 50² = 2500.

What is the Sum of First n Odd Natural Numbers?

The sum of first n odd natural numbers can be represented as 1 + 3 + 5 + ... + (2n - 1) where 1 is the first odd number and (2n - 1) represents the last odd number. There are n natural numbers in this AP series. The sum of n terms of an AP is given by the formula S_n= n/2 × [a + l], where a is the first odd number and l is the last odd number. Thus, the sum of the first n odd natural numbers is calculated as,
S_n = n/2 × [1 + (2n - 1)]
S_n= n/2 × [2n]
S_n= n × n
S_n= n²

What is the Mean of the Sum of n Odd Numbers?

The mean of the sum of n odd numbers is calculated as the sum of n odd numbers/number of terms = n²/n = n. Thus, the mean is n.