Let S _ n = Σ _ k = 1 ^ 4 n (- 1) k (k + 1) 2 k ^ 2 . Then S n can...

Given ${\text{S}}_{\text{n}} = \mathop{Σ}_{\text{k} = 1}^{4 \text{n}} \left(- 1\right) \dfrac{\text{k} \left(\text{k} + 1\right)}{2} {\text{k}}^{2}$
$= - {1}^{2} - {2}^{2} + {3}^{2} + {4}^{2} {- 5}^{2} - {6}^{2} + {7}^{2} + {8}^{2} + ............$
$= \left({3}^{2} - 1\right) + \left({4}^{2} - {2}^{2}\right) + \left({7}^{2} - {5}^{2}\right) + \left({8}^{2} - {6}^{2}\right) ............$
$= 2\left(1 + 3 + 2 + 4 + . . . + 4n\right)$
$= \dfrac{2\left(4n\left(4n + 1\right)\right)}{2} = 4n\left(4n + 1\right)$

${S}_{n} = 1056$ is true for $n = 8$

${S}_{n} = 1332$ is true for $n = 9$