Statement: Proof by induction for all $n\geq 1$
$$\begin{pmatrix}F_n\\F_{n+1}\end{pmatrix}=\begin{pmatrix}0 & 1\\\ 1 & 1\end{pmatrix}^{n-1}\begin{pmatrix}1\\1\end{pmatrix}$$
I'm know the induction steps but i keep having troubles with the use of induction in matrices. Im verified for n=1 and assumed n=k. But can't proced after assume that work for k+1.
I'm stuck in k+1 step.
$$\begin{pmatrix}F_{k+1}\\F_{k+2}\end{pmatrix}=\begin{pmatrix}0 & 1\\\ 1 & 1\end{pmatrix}^{k}\begin{pmatrix}1\\1\end{pmatrix}$$
I tried too
$$\begin{pmatrix}F_k+F_{k-1}\\F_{k+1}+F_k\end{pmatrix}=\begin{pmatrix}0 & 1\\\ 1 & 1\end{pmatrix}^{k}\begin{pmatrix}1\\1\end{pmatrix}$$
but idk how to procede
Because $\begin{pmatrix}0 & 1\\\ 1 & 1\end{pmatrix}\begin{pmatrix}a\\b\end{pmatrix} =\begin{pmatrix}b\\a+b\end{pmatrix} $.
(added later)
Therefore, and this is the induction step,
$\begin{pmatrix}0 & 1\\\ 1 & 1\end{pmatrix}\begin{pmatrix}F_n\\F_{n+1}\end{pmatrix} =\begin{pmatrix}F_{n+1}\\F_n+F_{n+1}\end{pmatrix} =\begin{pmatrix}F_{n+1}\\F_{n+2}\end{pmatrix} $.
This needs to be combined with
$\begin{pmatrix}0 & 1\\\ 1 & 1\end{pmatrix} \begin{pmatrix}0 & 1\\\ 1 & 1\end{pmatrix}^n =\begin{pmatrix}0 & 1\\\ 1 & 1\end{pmatrix}^{n+1} $.