`1^2 + 3^2 + 5^2 + (2n - 1)^2 = (n(4n^2 - 1))/3`  prove this using mathematical induction



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First check for `n=1`



So for `n=1` the identity holds. Now assume that the identity holds for all natural numbers greater or equal to `n` i.e.

`1^2+3^2+5^2+cdots+(2k-1)^2=(k(4k^2-1))/3,` `k leq n`             (1)

Let's now check for `k=n+1`


Now by using assumption (1) we get





which we were supposed to get. Hence our proof is done.

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