Answer
$\sum\limits_{k=1}^{8}{{{\left( -1 \right)}^{k+1}}{{2}^{k}}}$ is $-170.$
Work Step by Step
$\sum\limits_{k=1}^{8}{{{\left( -1 \right)}^{k+1}}{{2}^{k}}}$
For the sum of the notation,
$\begin{align}
& \sum\limits_{k=1}^{8}{{{\left( -1 \right)}^{k+1}}{{2}^{k}}}={{\left( -1 \right)}^{1+1}}{{2}^{1}}+{{\left( -1 \right)}^{2+1}}{{2}^{2}}+{{\left( -1 \right)}^{3+1}}{{2}^{3}}+{{\left( -1 \right)}^{4+1}}{{2}^{4}}{{\left( -1 \right)}^{5+1}}{{2}^{5}}{{\left( -1 \right)}^{6+1}}{{2}^{6}}{{\left( -1 \right)}^{7+1}}{{2}^{7}}{{\left( -1 \right)}^{8+1}}{{2}^{8}} \\
& =2-4+8-16+32-64+128-256 \\
& =-170
\end{align}$
Thus, the sum of the sigma notation $\sum\limits_{k=1}^{8}{{{\left( -1 \right)}^{k+1}}{{2}^{k}}}$ is $-170.$
