The sum of the first n odd natural numbers is n2.
Let's break down why this is the case and explore it further.
Understanding Odd Natural Numbers
Odd natural numbers are positive integers that are not divisible by 2. The first few odd natural numbers are 1, 3, 5, 7, 9, and so on. We can represent the nth odd natural number as 2n - 1.
The Arithmetic Progression
The sequence of odd natural numbers forms an arithmetic progression (AP). An arithmetic progression is a sequence where the difference between consecutive terms is constant. In this case, the common difference is 2 (e.g., 3 - 1 = 2, 5 - 3 = 2).
Deriving the Sum Formula
We can calculate the sum of the first n odd natural numbers using the arithmetic series formula. The sum (Sn) of an arithmetic series is given by:
Sn = (n/2) * [2a + (n-1)d]
Where:
- n = the number of terms
- a = the first term
- d = the common difference
In our case:
- a = 1 (the first odd natural number)
- d = 2 (the common difference between odd natural numbers)
Substituting these values into the formula, we get:
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 n2.
Examples
Here are a few examples to illustrate the formula:
- n = 1: The sum of the first 1 odd natural number (1) is 12 = 1.
- n = 2: The sum of the first 2 odd natural numbers (1 + 3) is 22 = 4.
- n = 3: The sum of the first 3 odd natural numbers (1 + 3 + 5) is 32 = 9.
- n = 4: The sum of the first 4 odd natural numbers (1 + 3 + 5 + 7) is 42 = 16.
Conclusion
In summary, the sum of the first 'n' odd natural numbers is equal to 'n' squared (n2). This concise formula provides a straightforward method for calculating the sum without having to add each individual number.
