Answer
$0$
Work Step by Step
$S_{1}=a_{1}=\left( -1\right) ^{1}=-1$
$S_{2}=a_{1}+a_{2}=\left( -1\right) ^{1}+\left( -1\right) ^{2}=0$
$S_{3}=a_{1}+a_{2}+a_{3}=\left( -1\right) ^{1}+\left( -1\right) ^{2}+\left( -1\right) ^{3}=-1$
$S_{4}=S_{3}+\left( -1\right) ^{4}=0$
We see if n is even $\Rightarrow S_{n}=0$
$100$ is even number, so
$S_{100}=\sum ^{100}_{j=1}\left( -1\right) ^{j}=0$
