Let-S-N-n-1-N-1-n-1-n-3-Find-S-2N-in-terms-of-N-

Question Number 1067 by Yugi last updated on 30/May/15

L e t

S_{N} = \underset{n = 1}{\sum^{N}} {(- 1)}^{n - 1} n^{3} .

F i n d S_{2 N} i n t e r m s o f N .

Answered by prakash jain last updated on 07/Jun/15

S_{N} = 1^{3} - 2^{3} + 3^{3} + \dots . . + N^{3}

S_{2 N} = \underset{i = 1}{\sum^{N}} {(2 i - 1)}^{3} - \underset{i = 1}{\sum^{N}} {(2 i)}^{3}

= \underset{i = 1}{\sum^{N}} 8 i^{3} - \underset{i = 1}{\sum^{N}} 1 - \underset{i = 1}{\sum^{N}} 12 i^{2} + \underset{i = 1}{\sum^{N}} 6 i - \underset{i = 1}{\sum^{N}} 8 i^{3}

= - N - 12 \frac{N (N + 1) (2 N + 1)}{6} - 6 (\frac{N (N + 1)}{2})

= N - 2 N (N + 1) (2 N + 1) - 3 N (N + 1)