Answer
The solution for the equations is $168$.
Work Step by Step
Consider the provided expression,
$\sum\limits_{i=1}^{6}{4{{\left( -2 \right)}^{i}}}$
Compare the provided expression with the standard form $\sum\limits_{i=1}^{n}{{{a}_{1}}{{\left( -r \right)}^{i}}}$.
So, $n=6$, ${{a}_{1}}=4$ and $r=2$.
Use the following sum expression for a geometric sequence,
${{S}_{n}}=\frac{{{a}_{1}}\left( 1-{{r}^{n}} \right)}{1-r}$
Substitute $n=6$, ${{a}_{1}}=4$ and $r=2$ in the expression ${{S}_{n}}=\frac{{{a}_{1}}\left( 1-{{r}^{n}} \right)}{1-r}$ ,
$\begin{align}
& {{S}_{6}}=\frac{{{a}_{1}}\left( 1-{{r}^{n}} \right)}{1-r} \\
& =\frac{4{{\left( -2 \right)}^{1}}\left[ 1-{{\left( 2 \right)}^{6}} \right]}{1-\left( -2 \right)} \\
& =168
\end{align}$
