Expectation for n'th Markov Chain Link

Question

I have a markov chain $X_n$ and have computed $a(n)$ such that $P(X_{n+1} = X_n + 1) = a(X_n)$, $P(X_{n+1} = X_n) = 1-a(X_n)$, with no other possible values for $X_{n+1}$. I have $X_0$ so this should allow me to calculate $E[X_n]$, but the only way I can think of doing so would be to build a transition matrix with $a(X_n)-1$ on the diagonals, $a(X_n)$ right above those entries and $0s$ everywhere else, and then raise it to the $n'th$ power, and manually calculate the expectation from the resulting distribution. The issue is that the state space is not finite, so the matrix would be infinitely large. Could someone help me out? Thanks.