Solution
= Solution
{c}
The total eventual mass of the closed class $\{4,5\}$ is $\lambda_4+\lambda_5+(1-p)\lambda_1$. The other closed class cannot contribute to state five. Since $|1-a-b|<1$, the two-state chain within the last class converges to its <stationary distribution>, with state-five mass $a/(a+b)$. Therefore
$$
\boxed{\lim_{n\to\infty}\mathbb P(X_n=5)
=\frac a{a+b}\bigl[\lambda_4+\lambda_5+(1-p)\lambda_1\bigr].}
$$