 Solution

The vertices visited by a simple symmetric random walk form an integer interval: nearest-neighbour steps cannot skip a vertex. At the stopping time $T_k$, write that interval as $[\ell ,r]$, with $r-\ell +1=k$. The walk is at one of its endpoints, because the $k$th new vertex must extend the earlier interval. This also covers $k=1$, when both endpoints coincide.

The Strong Markov property restarts the walk from this endpoint. Visiting a new vertex is exactly exiting $[\ell ,r]$, or reaching $\ell -1$ or $r+1$. Translate these absorbing boundaries to $0,k+1$. The starting point becomes either $1$ or $k$, so the expected duration of symmetric gambler's ruin is $k$ in both cases. Taking conditional expectations and then expected values gives the expected time to expand a random-walk range: