Answer
$8$
Work Step by Step
$\sum ^{8}_{i=1}\left[ 1+\left( -1\right) ^{i}\right] =\sum ^{8}_{i=1}1+\sum ^{8}_{i=1}\left( -1\right) ^{i}=8+\sum ^{8}_{i=1}\left( -1\right) ^{i}$
$\sum ^{1}_{i=1}\left( -1\right) ^{i}=\left( -1\right) ^{1}=-1$
$\sum ^{2}_{i=1}\left( -1\right) ^{i}=\left( -1\right) ^{1}+\left( -1\right) ^{2}=0$
$\sum ^{3}_{i=1}\left( -1\right) ^{i}=\left( -1\right) ^{1}+\left( -1\right) ^{2}+\left( -1\right) ^{3}=-1$
$\sum ^{4}_{i=1}\left( -1\right) ^{i}=\left( -1\right) ^{1}+\left( -1\right) ^{2}+\left( -1\right) ^{3}+\left( -1\right) ^{4}=0$
$
\Rightarrow \sum ^{2n}_{i=1}\left( -1\right) ^{i}=0\Rightarrow \sum ^{8}_{i=1}\left( -1\right) ^{i}=0$
$$\sum ^{8}_{i=1}\left[ 1+\left( -1\right) ^{i}\right] =\sum ^{8}_{i=1}1+\sum ^{8}_{i=1}\left( -1\right) ^{i}=8+\sum ^{8}_{i=1}\left( -1\right) ^{i}=8+0=8$$
