# A golden number

Probability Level 3

Suppose that you have a sequence of the type $\{F_k\}_{k \geq 0},$ that obeys the rule :

$F_n = F_{n-1}+F_{n-2} \quad \forall n \geq 2.$

Suppose that $F_0 , F_1 \in \mathbb R_0^{+}$ are given. Let $\phi_n = \frac{F_n}{F_{n-1}}$, $\ \ \ n >0$.
What is

$\phi = \lim_{n \rightarrow \infty} \phi_n?$

NOTE : In regards to some complains I have had, I would like to stress that : $\forall n>0, \ \phi_n >0$.