Let $\displaystyle \phi = \frac{1+\sqrt{5}}{2}$ and $\displaystyle \psi = \frac{1+\sqrt{5}}{2}$. Consider the Fibonacci sequence defined by:

$$ \displaystyle a_n = \frac{\phi^n - \psi^n}{\sqrt{5}} $$

Evaluate $\displaystyle \lim_{n \to \infty} \frac{\log(a_n)}{n}$.

My feeling is that I should work with the sequence $\displaystyle a_n^{1/n}$ and then $\log$ the result but I may be completely off. Any suggestions on how to go about this?