(1)+(1+3)+(1+3+5)+......+(1+3+5+...+21)?

1 Answer
Nov 12, 2017

Sum is 506

Explanation:

The given series (1)+(1+3)+(1+3+5)+....+(1+3+5+...+21) is equivalent to

1+4+9+.....+121 or 1^2+2^2+3^2+...+11^2

hence, this is sum up to 11^(th) term of series sumn^2.

Sum up to n terms of series sumn^2=(n(n+1)(2n+1))/6

hence desired sum is (11xx12xx23)/6=506

However to prove sumn^2=(n(n+1)(2n+1))/6, one can use induction method or a much more laborious proof is given here.