If the sum of $n$ terms of $2 + 4 + 8 + 16 + \cdots$ is $254$, then $n$ equals —

This is a geometric series with $a=2,\ r=2$, so $S_n=2(2^n-1)=2^{n+1}-2$. Setting $2^{n+1}-2=254$ gives $2^{n+1}=256=2^8$, hence $n=7$.