Suppose you have a random walk on $\{0,\ldots,N\}$, and let $v_k= E[T|x_o=k]$ be the mean hitting time starting in state $k$. Using fist step analysis we can easily see that $v_k$ satisfies the equation: $$v_k=1+q_kv_{k-1}+r_kv_k+p_kv_{k+1}$$ for $k=1,\ldots,N-1$ and $v_0=0=v_N$.
Using the fact that $q_k+p_k+r_k=1$, we can transform these equation to: $$v_k=1+q_kv_{k-1}+(1-p_k-q_k)v_k+p_kv_{k+1} $$ and then: $$ v_{k+1}-v_k=-1/p_k +\frac{q_k}{p_k}(v_k-v_{k-1}) $$ Iterating this formula, and defining $\Delta_k= v_{k+1}-v_k$ we get to: $$ \Delta_k= v_{k+1}-v_k = \prod_{j=1}^n \frac{p_j}{q_j}\Delta_0- \frac{1}{p_k}-\frac{q_k}{p_kp_{k-1}}-\frac{q_kq_{k-1}}{p_kp_{k-1}p_{k-2}}-\ldots -\frac{q_k\cdots q_2}{p_k\cdots_1}$$
Is this development correct? How can I solve for $v_k$?
Note: This is exercise 3.6.2 of Introduction to Stochastic Modelling, Pinsky & Karlin.