summation

1²+3²+5²+...........to n terms

2 Answers

not a very tough problem, but no one seems to have replied yet!

TO FIND:- 1^2+3^2+5^2+â€¦â€¦.to n terms. --------(S1)
nth term= (2n-1)^2

Nowâ€¦let S=1^2+2^2+3^2+â€¦â€¦..+(2n)^2
= 2n(2n+1)(4n+1)/6
= n(2n+1)(4n+1)/3

let S2= 2^2+4^2+6^2+â€¦â€¦â€¦..+(2n)^2
= 2^2(1^2+2^2+3^2+â€¦â€¦..+n^2)
=4n(n+1)(2n+1)/6
=2n(n+1)(2n+1)/3

so, S=S1-S2
=n(4n^2-1)/3 ------answer.

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