Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 2 / 20H / a / Solution

Until the walk first reaches $B$, it is the simple random walk on a path of length $n$ with reflection at the endpoint $A$. If $h_i$ is the expected hitting time of $B$ from the vertex at distance $i$ from $A$, thenSolving this second-difference equation gives $h_i=n^2-i^2$. Therefore