For 1D random walk starting at 0 with probability $p$ up one step and $q$ down one step, let $T_k$ be the random variable that counts the number of steps for a path to first reach $k$.
It feels like $T_2-T_1$ should be identically distributed with $T_1$, but this doesn't seem to be true. I got $P(T_2-T_1=t)=rP(T_1=t)$, where $r=\sum_{n=0}^\infty P(T_1=n)$, i.e. $r$ is the probability of reaching 1 eventually (which is smaller than 1 if $p<0.5$). What did I do wrong?