A golden number

Probability Level 3

Suppose that you have a sequence of the type {Fk}k0,\{F_k\}_{k \geq 0}, that obeys the rule :

Fn=Fn1+Fn2n2. F_n = F_{n-1}+F_{n-2} \quad \forall n \geq 2.

Suppose that F0,F1R0+ F_0 , F_1 \in \mathbb R_0^{+} are given. Let ϕn=FnFn1\phi_n = \frac{F_n}{F_{n-1}},    n>0\ \ \ n >0.
What is

ϕ=limnϕn? \phi = \lim_{n \rightarrow \infty} \phi_n?

Rounded your answer to the nearest thousandths.

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


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