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 $n=\boxed{8}$.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
The sum of the first $n$ terms in the infinite geometric sequence $\left\{\frac{1}{4},\frac{1}{8},\frac{1}{16},\dots \right\}$ is $\frac{255}{512}$. Find $n$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
8
The sum of the first $n$ terms in the infinite geometric sequence $\left\{\frac{1}{4},\frac{1}{8},\frac{1}{16},\dots \right\}$ is $\frac{255}{512}$. Find $n$.
test