Compute \(F_{n+1}F_{n-1} - F_n^2\) for a few values of \(n\); guess a general formula and prove it by induction, or otherwise.

Are we sure we know what proof by induction looks like?

Could we assume our general formula holds for \(F_{n+1}F_{n-1} - F_n^2\) and then consider \(F_{n+2}F_{n} - F_{n+1}^2\)?

By induction on \(k\), or otherwise, show that \[F_{n+k} = F_k F_{n+1} + F_{k-1} F_n\] for all positive integers \(n\) and \(k\).

Can we use induction on \(k\) without needing to use induction on \(n\)?