Markov Chains & Long-Run Behavior · Challenge problem

A two-state chain has transition matrix $M = \begin{pmatrix} 3/4 & 1/4 \\ 1/4 & 3/4 \end{pmatrix}$ and starts in state 1. Let $p_n$ be the probability that it is in state 1 after $n$ steps. What is the smallest $n$ with $p_n < 0.501$?