One-dimension random walk expected exit time

Question

There is a pawn in $x=0$ at $t=0$. At $t'=t+1$ the pawn moves to $x+1$ or $x-1$ with probabilities $p$, $1-p$.
What is the mean time to escape from the boundaries $-k$, $+n$, where $k,n\in \mathbb N$, $k,n>0$ ? I know the answer is $kn$ for $p=\frac12$, I would like to know how to prove it and how to find and prove the answer for arbitrary $p\in (0,1)$.
Thank you.

Answer 1

Solving the functional equation I have $$E=\dfrac{k\,p^{n+k}+n\,q^{n+k}-(n+k)\,p^kq^n}{(p-q)(p^{n+k}-q^{n+k})}$$where $q=1-p$.
The formula has no meaning for $p=q=\frac12$ but we have $$\lim_{p\to\frac12}E=kn$$as it should be.