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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

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