Let $X_n$ be a Markov chain with states $S=\{0, ..., N\}$ and transition probabilities $p(x,y)=p\mathbb{1}_{y=x+1}+(1-p)\mathbb{1}_{y=0}$, where $0\le x \le N-1$ and $0 \lt p \lt 1$. Let $T=\inf\{n\ge 0 : X_n = N\}$. Compute $u(x):=E[T|X_0=x]$
So we have that $u(x)=0$ if $x=N$ and from the Markov property we get $u(x)=1+\sum_{y\in S}p(x,y)u(y)=1+(1-p)u(0)+pu(x+1)$ for $x\ne N$. Now I am stuck, since I seem to need a second boundary condition, I am not able to resolve this recurrence relation to a direct formula using just $u(N)=0$. I would appreciate any hints.