Proof that the Golden Ratio is Approached by the Ratio of Adjacent Terms of the Fibonnaci Sequence


Kimberly Burrell

Recall that the Fibonnaci Sequence is the sequence described by f(0)=1, f(1)=1, with f(n)=f(n-1) + f(n-2). In particular, we are interested in the ratio f(n+1)/f(n). We will call the ratio


We can assume that the limit of x sub n as n approaches infinity is x.


or .

Since, we are dealing with only the positive values of x,


which equals the Golden Ratio 1.618033988738303.