The Fibonacci numbers \(F_n\) are defined by the conditions \(F_0 = 0\), \(F_1 = 1\) and \[F_{n+1} = F_n + F_{n-1}\] for all \(n \geq 1\). Show that \(F_2 = 1\), \(F_3 = 2\), \(F_4 = 3\) and compute \(F_5\), \(F_6\) and \(F_7\).
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.
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 \(k\).
