Mathematics - Sequence & Series Question with Solution | TestHub

Given ${\text{S}}_{\text{n}}=\underset{\text{k}=1}{\overset{4\text{n}}{\Sigma }}{\left(-1\right)}^{\frac{\text{k}\left(\text{k}+1\right)}{2}}{\text{k}}^2$
$=-1^2-2^2+3^2+4^2{-5}^2-6^2+7^2+8^2+............$
$=\left(3^2-1\right)+\left(4^2-2^2\right)+\left(7^2-5^2\right)+\left(8^2-6^2\right)............$
$=2\left(1+3+2+4+...+4n\right)$
$=\frac{2\left(4n\left(4n+1\right)\right)}{2}=4n\left(4n+1\right)$

$S_n=1056$ is true for $n=8$

$S_n=1332$ is true  for $n=9$