jmc

This is a geometric sequence with first term $\frac{1}{4}$ and common ratio $\frac{1}{2}$. Thus the sum of the first $n$ terms is:

$\frac{255}{512}=\frac{1}{4}\left(\frac{1-{\left(\frac{1}{2}\right)}^n}{1-\frac{1}{2}}\right)=\frac{2^n-1}{2^{n+1}}$.

We see that $\frac{255}{512}=\frac{2^8-1}{2^9}$, so $n=\boxed{8}$.