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*).

I had trouble proving this using normal analysis, but it follows simply if using the closed discrete formula for F(n)

F(n) = (Phi^n - (-1/Phi)^n)/sqrt(5)

Then R(n) = (Phi^n - (-1/Phi)^n)/(Phi^(n-1) - (-1/Phi)^(n-1))

As n goes to infinity, the (-1/Phi)^n terms go to zero, so

the limit of R(n) as n goes to infinity exists and equals the limit as n goes to infinity of Phi^n/(Phi^(n-1), which is Phi