I have a standard proof for the theorem:
$$\sum_{}^n f_1+f_3+f_5+...+f_{2n-1} = f_{2n}$$
$$f_i$$ refers to the Fibonacci numbers for future reference.
It involves setting p(k) as p(k+1) and proving it through weak induction, it has been graded by my professor and is correct.
However, I have recently came across the fibonacci matrix formulation from here:
How to prove Fibonacci sequence with matrices?
I am curious how I would go about solving this theorem with matrices.
I have tried using the product operator: $$\Pi$$
but I am not experienced enough to correctly formulate it so that they equal each other, for example:
$$\Pi_{i=1}^n \begin{bmatrix}1 & 1\\1 & 0\end{bmatrix}^{2n-1} = \begin{bmatrix}1 & 1\\1 & 0\end{bmatrix}^{2n}$$
Using the product operator may be completely pointless, but I honestly just don't know since I have never really used them before.
Any idea on how the original theorem is shown in matrices?
Thank you for any help in advance.