TITLE: Product of Fibonacci numbers
QUESTION [6 upvotes]: I'm looking for the asymptotic approximation of the product of the first $n$ Fibonacci numbers.
Does there exist a tight approximation for these kind of things?

REPLY [16 votes]: By the Binet formula,
$$F_n\approx\frac{\phi^n}{\sqrt5}.$$
Then multiplying the $n$ first estimates
$$P_n\approx a\frac{\phi^{n(n+1)/2}}{\sqrt5^n}.$$
By numerical computation, $a\approx 1.22674201072$.

We can deduce an expression for the geometric average
$$\sqrt[n]P_n\approx\frac{\phi^{(n+1)/2}}{\sqrt5}\approx\frac{F_n}{\sqrt{5\phi}}.$$