Ratios of adjacent terms of Fibonacci and Lucas Sequences

Fibonacci and Lucas sequences are of the form fn+1 = fn + fn-1. Thus, the following ratios are equivalent:

fn+1/fn = (fn + fn-1)/fn

or

fn+1/fn = 1 + fn-1/fn

 

Let the limit of fn+1/fn = L as n approaches infinity. Equivalently, the limit of fn/fn-1 also equals L. Thus, the previous equation becomes:

(golden ratio)

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