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Recurrence Relationshard
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Let FnF_nFn​ denote the Fibonacci sequence with F0=0,F1=1F_0 = 0, F_1 = 1F0​=0,F1​=1, satisfying Fn=Fn−1+Fn−2F_n = F_{n-1} + F_{n-2}Fn​=Fn−1​+Fn−2​. The characteristic roots are ϕ=1+52\phi = \frac{1+\sqrt{5}}{2}ϕ=21+5​​ and ϕ^=1−52\hat{\phi} = \frac{1-\sqrt{5}}{2}ϕ^​=21−5​​. Which of the following equals lim⁡n→∞Fn+1Fn\lim_{n \to \infty} \frac{F_{n+1}}{F_n}limn→∞​Fn​Fn+1​​?