Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics III / 5 / b / ii

Let $q_n=\mathbb{P}(M_n=Y)$. The law of total probability applied to the current markov chain state givesThe second equality holds because $Y$ can only be reached from $X$. Eliminating $q_{n-1}$ gives the linear recurrence relation