I have been studying markov chains for my Introductory Stochastic Processes exam, but i am struggling with the following problem:
Question: Consider a matrix with state space $S=\{1,2,3\}$ and the following one step transition matrix: $$P=\left(\begin{array}{ccc}0 & 0.5 & 0.5 \\ 0.5 & 0 & 0.5 \\ 0 & 0 & 1\end{array}\right)$$
Let $X_{n}$ be the state of the chain at time $n$ and suppose that $X_{0} = 1$. Find the probability that $\mathbb{P}\left(X_{n}=2\right)$ for every $n \in \mathbb{N}$.
What have i tried so far? I tried writing $$P\left(X_{n}=j \mid X_{0}=i\right)= P\left(X_{n}=2 \mid X_{0}=1\right) = \left(P^{n}\right)_{12}$$ since we know that for time-homogeneous, discrete markov chains, the following holds: $$ P_{i j}^{n}=P\left(X_{n}=j \mid X_{0}=i\right), \text { for all } i, j $$ However, when raising the transition matrix $P$ to the nth power, entries behave differently depending on the parity of n, which got me a bit confused. Is my approach solid? Is there another way of solving this problem?
Any help is appreciated. Thanks in advance, Lucas