Question

Prove that the sum of the first n odd natural numbers are equal to n^

Answer 1

Answer:

Proved below.

Step-by-step explanation:

We know that

The sum of first n terms in an A.P with common difference d is

Sₙ = n/2 (2a + (n - 1) d)               where a is the first term

First n odd natural numbers = {1,3,5,7,...............,n}

⇒ a = 1

⇒d = a₂ - a₁ = 3 - 1 = 2

∴ Sum of first n odd natural numbers = n/2 [2a + (n - 1) d]

                                                              = n/2 [2(1) + (n - 1)2]

                                                              = n/2 [2 + 2n - 2]

                                                              = n/2 (2n)

                                                              = n²