The sum of the first n odd natural numbers is n2.
Understanding the Sum of Odd Natural Numbers
The sequence of odd natural numbers forms an arithmetic progression: 1, 3, 5, 7, 9,...
- First Term (a): 1
- Common Difference (d): 2 (since each term is 2 greater than the previous)
The nth odd natural number can be expressed as 2n - 1. For instance, the 1st odd number is (21)-1 = 1, the 2nd is (22)-1 = 3, and so on.
Proving the Formula: Sum of n Odd Numbers = n2
We can prove this using the formula for the sum of an arithmetic series:
Sn = (n/2) * [2a + (n-1)d]
Where:
- Sn is the sum of the first n terms
- n is the number of terms
- a is the first term
- d is the common difference
In our case, a = 1 and d = 2. Substituting these values:
Sn = (n/2) [2(1) + (n-1)2]
Sn = (n/2) [2 + 2n - 2]
Sn = (n/2) * [2n]
Sn = n2
Therefore, the sum of the first n odd natural numbers is indeed n2.
Examples
Here are a few examples to illustrate the formula:
- Sum of the first 1 odd number: 1 = 12
- Sum of the first 2 odd numbers: 1 + 3 = 4 = 22
- Sum of the first 3 odd numbers: 1 + 3 + 5 = 9 = 32
- Sum of the first 4 odd numbers: 1 + 3 + 5 + 7 = 16 = 42
- Sum of the first 5 odd numbers: 1 + 3 + 5 + 7 + 9 = 25 = 52
As you can see, the formula holds true for these examples.
