Consider a Markov chain with state space $$S = \{1, 2, \dots, 4, 5\}$$ and transition matrix $$P = \begin{pmatrix} 0 & 0 & \frac 1 2 & \frac 1 2 & 0\\ \frac 1 2 & 0 & \frac 1 2 & 0 & 0\\ \frac 1 2 & 0 & 0 & \frac 1 2 & 0\\ 0 & \frac 1 2 & 0 & 0 & \frac 1 2\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix}.$$ Let $$k_i = \mathbb{E}_i(T_5),$$ where $$T_y = \min\{n \geq 1 : X_n = y\}\ \forall\ y \in S.$$ By using a first step analysis, determine the corresponding equations for the $k_i$ as well as the numerical value of $k_1$.
My working
$$\begin{aligned} k_1 & = \sum^5_{x = 0} \mathbb{E}_1(T_5 \mid X_1 = x)\mathbb{P}_1(X_1 = x)\\[1 mm] & = (1 + k_3)\frac 1 2 + (1 + k_4)\frac 1 2\\[1 mm] & = 1 + \frac 1 2 k_3 + \frac 1 2 k_4 \end{aligned}$$ Similarly, we have $$k_2 = 1 + \frac 1 2 k_1 + \frac 1 2 k_3,$$ $$k_3 = 1 + \frac 1 2 k_1 + \frac 1 2 k_4$$ and $$k_4 = 1 + \frac 1 2 k_2.$$ Solving this, we have $$k_1 = 7.$$
As I am just beginning to learn about Markov chains, this is my first time encountering such a problem and I would like to know if my working and answer are correct. Otherwise, do kindly point out where I have gone wrong and why :)