 Solution

Starting at one, state two can first be reached only after an even number of steps. At every even time before that hit, the chain is again at one. By the Markov property and part (i), each successive two-step block hits two with probability $4/9$, while a failure returns to one with probability $5/9$. Therefore, for the first hitting time $H$,Equivalently $H/2$ has the geometric distribution on the positive integers with success probability $4/9$.