Maximum constant $c$ for which $F_n = \Omega(2^{cn})$, where $F_n$ is the Fibonacci sequence

Question

What is the maximum real constant $c$ for which $F_n = \Omega(2^{cn})$, where $F_n$ is the Fibonacci sequence?

I figured out that I have to calculate the maximum $c$ for which the limit $\lim_{n \to \infty}\frac{2^{cn}}{F_n}$ is finite. I tried writing $F_n$ with the explicit formula but I still couldn't figure it out. This is actually from problem 0.3, subproblem c) in Dasgupta's book Algorithms.

Answer 1

Since $F_n$ is proportional to $\phi^n$, $2^{cn}$ must be proportional to $x^n$ where $x ≤ \phi$. Therefore, the maximum value of $c$ achieves $2^c = \phi$, so it is $\log_2(\phi)$.