Is there some sort of trick when dealing with the following sum?:
$\displaystyle \sum_{k=2} ^{99} (-1)^k(k)^2$
I attempted it numerous times, but the (-1)^k is messing me up
We can start from the known formula...
$\displaystyle \displaystyle S^{2}_{n} = \sum_{k=1}^{n} k^{2} = \frac{n\ (n+1)\ (2n+1)}{6}$ (1)
... and from (1) derive with some 'easy' step...
$\displaystyle \displaystyle \sum_{k=2}^{99} (-1)^{k} k^{2} = 1-S^{2}_{99} + 8\ S^{2}_{49} =- 4949$ (2)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$