Generalized Fibonacci Sequence and the Golden Ratio

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The sequence of Fibonacci numbers is given by 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …, in which each number is the sum of the two preceding numbers. As , the ratio approaches , known as the golden ratio (or golden section or divine proportion), designated by .

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A remarkable generalization of this result is that for an arbitrary pair of numbers and (not both zero), the generalized Fibonacci sequence gives the same limiting ratio of successive members as , independent of the choices of and .

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Contributed by: S. M. Blinder (July 2018)
Open content licensed under CC BY-NC-SA


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Details

Consider the original Fibonacci sequence: divide the recursion relation by to give

.

With some work, you can prove that the ratios form a Cauchy sequence, so the sequence has a limit; call it . Thus

while

.

This gives or , a quadratic equation with roots

.

The positive root

gives the golden ratio.

For the generalized Fibonacci sequence , the general term can be written using Fibonacci numbers as . Therefore,

.

In the limit , this approaches

,

independent of or , as long as not both are zero.



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