## Alternate 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 $F_2 = 1$.

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

Consider the ratio of adjacent terms, $\dfrac{F_{n+1}}{F_{n}}$:

1/1 = 1

2/1 = 2

3/2 = 1.5

5/3 = 1.666...

8/5 = 1.6

13/8 = 1.625

21/13 = 1.615...

34/21 = 1.619...

55/34 = 1.617...

89/55 = 1.618...

As $n$ increases it is well known that the ratio of adjacent terms $\dfrac{F_{n+1}}{F_{n}}$ tends towards $\phi = \dfrac{\sqrt{5} + 1}{2} \approx 1.618$ (see Fibonacci Ratio).

What does the ratio $\dfrac{F_{n+2}}{F_n}$ tend towards as $n$ increases?

Problem ID: 310 (15 Feb 2007) Difficulty: 2 Star