Review question

# Can we prove these Fibonacci number results? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9868

## Suggestion

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 $n$?