Sum of n Odd Numbers Formula
The sum of odd numbers can be expressed using a formula. If you want to find the sum of the first n odd numbers, the formula is:
S = n2
Where,
- S is the sum of the first n odd numbers, and
- n is the number of odd numbers you want to add.
Proof of Sum of Odd Numbers Formula
The odd numbers are 1, 3, 5, 7, 9, 11, . . . from the header above, as can be seen. If students examine closely, they can find an arithmetic progression sequence (AP). The AP can be included into the formula in the following ways:
In first step, Lets see the simple formula of Sum of Odd numbers:
(2n±1) It can represented as 2n+1 or 2n- 1
- In 2n+1 it can be begins with n= 0, 1, 3, 5, 7, 9 or so on.
- In 2n- 1 it can be begins with n= 1, 3, 5, 7, 9 or so on.
- For Ex- n= 0, 2n+1 becomes 1 or n= 1, 2n- 1 than also it becomes 1.
- It can be written as Sn = 1 + 3 + 5 + 7 + … + (2n−1).
In Second Step, Lets see the AP Formula:
Sn = n/2×(2a+(n−1)d)
- n is the number of terms,
- a is the first term
- d is the common difference
In Third Step, How we apply AP in Odd numbers:
- Here, a is the first term is 1
- And d is the common difference is 2.
- Even, the last term L is 2n- 1.
At last Substituting values of AP:
Sn = (n/2) × (1 + 2n – 1)
After Simplifying we get: Sn= (n/2) × (2n) = n2
So, Sum of Odd numbers in each terms is n2.
Sum of Odd Numbers
Sum of Odd Numbers is calculated by adding together integers that are not divisible by 2, resulting in a total that is either an odd number or even number. Sum of Odd Numbers is often represented by the formula expressed as n2 where n is a natural number. This formula can be used to calculate the sum of the first n odd numbers without adding them individually.
In this article, we will learn about the Sum of Odd Number Formula including the definition of Odd Numbers as well as some solved examples using the formula.
Table of Content
- What are Odd Numbers?
- How to Find the Sum of Odd Numbers?
- Sum of n Odd Numbers Formula
- Sum of Odd Numbers from 1 to 100
- Solved Examples
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