Expected time for an upward-biased walk to reach 7

A particle on the integers starts at $0$ and at each step moves $+1$ with probability $\tfrac23$ or $-1$ with probability $\tfrac13$. Find the expected number of steps until it first reaches $7$ (or say if this is infinite).

$21$. Reaching 7 means seven successive first-passages up by one level, each with the same mean. For one level, $e=1+\tfrac13\cdot2e$ gives $e=3$ (equivalently, distance divided by the drift $\tfrac13$). So the answer is $7\cdot3=21$.