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

## Question

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 $k$.