Show by the Principle of Mathematical induction that the sum `S_n`, of the nterms of the series `1^2 + 2xx 2^2 + 3^2 + 2xx 4^2+5^2 +2 xx 6^2 +7^2+..... ` is given by `S_n={(n(n+1)^2)/2`, if n is even , then `(n^2(n+1))/2` , if n is odd

Explanation

`S_n=1^2+2×2^2+3^2+2×4^2+.`

when n=2,

`S_2=1^2+2×2^2=1+8=9`


From RHS, we have if n is even `S_n=(n(n+1)^2)/2
S_2=(2×9)/2=9`

Let assume above is true for n = k we get

k is even,
`S_k=1^2+2×2^2+3^2+2×4^2+....+2×k^2 ........(i)`

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