Answer
$\sum_{k=1}^{100}\left(-2^{k}+2^{1+k}\right)=-2+2^{101}$
Work Step by Step
We find:
$\sum_{k=1}^{100}\left(-2^{k}+2^{k+1}\right)=\left(2^{2}-2^{1}\right)+\left(2^{3}-2^{2}\right)+\left(2^{4}-2^{3}\right)+\ldots+\left(2^{101}-2^{100}\right)$
$=\left(2^{2}-2\right)+\left(2^{8}-2^{2}\right)+\left(2^{1}-2^{8}\right)+\ldots+\left(2^{101}-2^{100}\right)$
$=-2+2^{101}$
$\therefore \sum_{k=1}^{100}\left(-2^{k}+2^{1+k}\right)=-2+2^{101}$
