Question

Help with indicated sum question?

Answer 1

Since \((-1)^k=1\) for even \(k\) and \(-1\) for odd \(k\), you can split up the sum into even and odd cases, then simplify:
\[\begin{align*}
\sum_{k=1}^4(-1)^k(k+11)&=\sum_{\ell=1}^2(-1)^{2\ell}(2\ell+11)+\sum_{m=1}^2(-1)^{2m-1}(2m-1+11)\\[1ex]
&=\sum_{\ell=1}^2(2\ell+11)-\sum_{m=1}^2(2m+10)\\[1ex]
&=\sum_{\ell=1}^21\\[1ex]
&=2
\end{align*}\]