I am asked to compute the stationary distribution of the markov chain with state space $E=\{0\dots,n\}$ and transition matrix below:
\begin{bmatrix} 0 & 1 \\ \frac{1}{n} & 0 & \frac{n-1}{n} \\ & \frac{2}{n} & 0 & \frac{n-2}{n} & \\ & & \ddots & \ddots & \ddots \\ & & & \frac{n-1}{n} & 0 & \frac{1}{n} \\ & & & & 1 & 0 & \\ \end{bmatrix}
What I did was use $\pi P = \pi$. And I got:
$\pi_0=\frac{1}{n}\pi_1 \Rightarrow \pi_1 =n\pi_0 \\ \pi_1 =\pi_0 +\frac{2}{n}\pi_2 \Rightarrow \pi_2 =\frac{n(n-1)}{2}\pi_0 \Rightarrow \pi_2=\frac{n-1}{2}\pi_1 \\$
I tried fiddling with it here and there but I cant seem to get anywhere to finish this problem. i.e. I can't seem to find $\pi_k$ for all $k \in E=\{0,\dots,n\}$. How would I finish this problem?