 Solution

Let $u(i)$ be the expected number of visits to state $3$ before absorption, counting the present state when $i=3$. Its reward equations areThe last equation gives $u(3)=2+u(2)$, so the second nontrivial equation gives $u(2)=2+u(1)$. Hence $u(1)=1+u(1)/2$, andThus the expected occupation count is two; together with the previous parts, this is the four-state reflecting absorbing random walk calculation.