 Solution

Let $m_k$ be the expected remaining hitting time of either endpoint, measured in minutes. The first step consumes one minute, after which first-step analysis givesThe expectation is finite: from any interior state, a block of twenty leftward steps guarantees absorption and has probability $2^{-20}$. Thus the survival probability over successive twenty-step blocks has a geometric upper bound. This justifies using finite expectations in the recurrence.