Consider the symmetric, simple random walk on $S = \{0, 1, \ldots , k\}$ for $k \in \mathbb N$.
Let $$T = \min \{ n \in \mathbb N_0|X_n = 0\}$$
be the first time where the process reaches $0$ and let $a_x = E\{T|X0 = x\}$ be the expected time for reaching $0$ when starting at $x \in S$.
ax satisfies $a_0 = 0$ and $$a_x =\frac12 a_{x−1} + \frac12 a_{x+1} + 1 \tag{1}$$ for all $x ∈ \{1, 2, . . . , k − 1\}$.
How do I find a relation similar to $(1)$ for $a_k$?
I am not sure what happens at the barrier of a simple symmetric random walk, does it go to k-1 next with probability $1$ or can it stay where it is too?
Solve the resulting system of equations to obtain the values $a_0, a_1, . . . , a_k$. I cannot do this (possibly because I do not know ak)