find the sum of the n terms of the series

Question

S_n  =1² +3² +5² .... + (2n-1)² .

Answer 1

This is a tough one.  There are a couple of proofs that can be given, but the simplest is by mathematical induction, and that requires knowing the answer first in order to do the proof.  Deriving the answer from scratch is difficult.

 

I don't usually like to just give answers without any explanation, so I won't give the answer here.  However, I will point out that each number in the series is the square of the nth odd number.  And I was able to find the answer by doing a web search for "sum of first n odd squares"