Let $\Phi$ be the golden ratio and $F_n$ be the usual Fibonacci numbers. How can I derive the following formula?
$$
\Phi = \lim_{n\rightarrow \infty} \sqrt[n]{F_n}
$$
I know the usual relation
$$
\Phi = \lim_{n\rightarrow \infty} \frac{F_{n+1}}{F_n} \quad ,
$$
and Wikipedia tells me that
$$
\Phi^a = \lim_{n\rightarrow \infty} \frac{F_{n+a}}{F_n} \quad .
$$
My first idea was to set $a = n$, which gives
$$
\Phi = \lim_{n\rightarrow \infty} \sqrt[n]\frac{F_{n+n}}{F_n} \quad ,
$$
EDIT:
We can also do
$$
\Phi = \lim_{n\rightarrow \infty} \sqrt[n]{\frac{F_{n+n}}{F_n}\frac{F_n}{F_n}} \quad ,
$$
but I am totally stuck here…
Best Answer
It is a standard result that $F_n = \frac{\phi^n - (-\phi)^{-n}}{\sqrt{5}}$.
Then $\lim_{n\rightarrow \infty} \sqrt[n]{F_n} = \lim_{n\rightarrow \infty} \sqrt[n]{\frac{\phi^n - (-\phi)^{-n}}{\sqrt{5}}} = \lim_{n\rightarrow \infty} \sqrt[n]{\frac{\phi^n}{\sqrt{5}}} = \lim_{n\rightarrow \infty} \frac{\phi}{\sqrt[n]{\sqrt{5}}} = \phi$ .