## Fibonacci Ratio

#### Problem

The Fibonacci sequence is defined by the second order recurrence relation $F_{n+2} = F_{n+1} + F_n$, where $F_1 = 1$ and $F_2 = 1$.

$1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...$

Assuming that ratio of adjacent terms in the Fibonacci sequence $\dfrac{F_{n+1}}{F_n}$ tends to a limit, $\phi$, as $n$ increases, prove that $\phi = \dfrac{\sqrt{5} + 1}{2}$.

Problem ID: 311 (15 Feb 2007) Difficulty: 3 Star