Proof that the Limit of the Ratio of

Adjacent terms of the Fibonnaci Sequence

is the Golden Ratio

by Amy Benson

Remember that the Fibonnaci Sequence is the sequence described by f(0)=1, f(1)=1, with f(n) = f(n-1) + f(n-2). We are interested in the the ratio f(n+1)/f(n). Let's call the the ratio

We can assume that the limit of xsub n as n approachs infinity is x.

So,

or

Since we are dealing only with positive values of x,

which equals the golden ratio 1.618033988738303.


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