In a Fibonacci sequence

**1, 1, 2, 3, 5, …, F**_{n}, F_{n+1}
define

**R**_{n} = F_{n}/ F_{n-1}
Prove that lim (R

_{n}) as n approaches infinity

is

**.5*(1+sqrt(5))=1.618...**

a.k.a.

**the golden ratio, φ (***phi*).

Suppose the required limit exits and = x.

Now F.n / F.n-1 = (F.n-1 + F.n-2) / F.n-1 by definition of F.n

= 1 + F.n-2 / F.n-1

Taking the limit as n --> infinity: x = 1 + 1/x

This results in a quadratic: x^2 - x - 1 = 0

By the familiar quadratic formula: x = (1 +- sqr(5) ) / 2

The 2 solutions are: 1.618... and -0.618....

QED!

x is the golden ratio, known as phi.