Find the eigenvalues and eigenvectors of a $n\times n$ Toeplitz matrices.
$\begin{bmatrix} a & c &0 &0&...&0&0 \\ b & a &c &0&...&0&0 \\ 0 & b &a&c&...&0&0 \\ \vdots & \vdots &\vdots &\vdots &\ddots&\vdots & \vdots\\ 0 & 0 &0&0 &\dots&b&a \end{bmatrix}$
(Hint:- Eigenvalues is of the form:- $\lambda_k=a+2\sqrt{bc}\cos(\frac{k\pi}{n+1}), k=1,2,3,..,n)$
My attempt:- I tried to apply induction on $n$.
For $n=1$. The result is true. Assume result true for $n=m$.
For the matrix $(n-1)\times(n-1)$
I made the block matrices. $\begin{bmatrix} a & c &0 &0&...&0&\vdots&0 \\ b & a &c &0&...&0&\vdots&0 \\ 0 & b &a&c&...&0&\vdots&0 \\ \vdots & \vdots &\vdots &\vdots &\ddots&\vdots &\vdots& \vdots\\ \dots & \dots &\dots &\dots &\dots&\dots &\vdots& \dots\\ 0 & 0 &0&0 &\dots& b &\vdots& &a \end{bmatrix}$
I am not able to proceed further. Could you give some help?