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

**by**

**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.**

**Thus,**

** or .**

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

**,**

**which equals the Golden Ratio
1.618033988738303.**