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.

Posted by JayDeeKay on 2015-09-16 09:17:40