Prove that the sum of the first n odd natural numbers is n2

Prove that the sum of the first n odd natural numbers is n^2. (Induction on n.) Let S(n) denote the sum of the first n odd natural numbers. Note that the nth odd natural number is 2n - 1, for n greaterthanorequalto 1. S(1) = 1 = 1^2. Suppose as inductive hypothesis that S(k - 1) = (k - 1)^2 for some k > 1. Then so, by induction, S(n) = n^2 for all n greaterthanorequalto 1.

Solution

Need to prove that the sum of first n odd natural numbers is n².

From the given information,

s(k) = s(k-1)+2k-1

= (k-1)² + 2k-1

= k²-2k+1+2k-1

= k²

Hence proved that s(k) = k²

So, by induction, S(n) = n² for all n>=1