Determine the index of a given Fibonacci number
Solution 1:
From Binet’s formula we have
$\sqrt{5}F_n=\phi^n-(-\phi)^{-n}$
and since $\phi^2 = \left(\frac{1+\sqrt5}{2}\right)^2 = \frac{1+2\sqrt5+5}{4} > \frac{10}{4} > 2$ we have
$\sqrt{5}F_n + \frac12 > \phi^n \space \forall n\ge 2$
But $\sqrt{5}F_n$ cannot be as great as $\phi^{n+1}$ so
$n<\log_{\phi}(\sqrt{5}F_n+\frac12)<n+1 \space \forall n \ge2$