Answer
$8-16+32-64+\ldots+(-1)^{n+1}2^{n}$
Work Step by Step
There are n-2 terms, as the index k changes from 3 to n.
The index k dictates how the terms are formed:
$\displaystyle \sum_{k=3}^{n}(-1)^{k+1}2^{k}=$
$=(-1)^{3+1}2^{3}+(-1)^{4+1}2^{4}+(-1)^{5+1}2^{5}+...+(-1)^{n+1}2^{n}$
$=(-1)^{4}2^{3}+(-1)^{5}2^{4}+(-1)^{6}2^{5}+\cdots+(-1)^{n+1}2^{n}$
When the exponent of $(-1)^{n}$ is odd, we have $-1$
and when it is even, $+1.$
$=2^{3}-2^{4}+2^{5}-2^{6}+\cdots+(-1)^{n+1}2^{n}$
$=8-16+32-64+\ldots+(-1)^{n+1}2^{n}$
