Answer
$\sum\limits_{k=1}^{7}{{{\left( -1 \right)}^{k}}{{4}^{k+1}}}$ is $-52,432\text{ }\text{.}$
Work Step by Step
$\sum\limits_{k=1}^{7}{{{\left( -1 \right)}^{k}}{{4}^{k+1}}}$
For the sum of the notation,
$\begin{align}
& \sum\limits_{k=1}^{7}{{{\left( -1 \right)}^{k}}{{4}^{k+1}}}={{\left( -1 \right)}^{1}}{{\left( 4 \right)}^{1+1}}+{{\left( -1 \right)}^{2}}{{\left( 4 \right)}^{2+1}}+{{\left( -1 \right)}^{3}}{{\left( 4 \right)}^{3+1}}+{{\left( -1 \right)}^{4}}{{\left( 4 \right)}^{4+1}}+{{\left( -1 \right)}^{5}}{{\left( 4 \right)}^{5+1}}+{{\left( -1 \right)}^{6}}{{\left( 4 \right)}^{6+1}}+{{\left( -1 \right)}^{7}}{{\left( 4 \right)}^{7+1}} \\
& =-16+64-256+1,024-4,096+16,384-65,536 \\
& =-52,432
\end{align}$
Thus, the sum of the sigma notation $\sum\limits_{k=1}^{7}{{{\left( -1 \right)}^{k}}{{4}^{k+1}}}$ is $-52,432\text{ }\text{.}$
