# Product of Fibonacci numbers

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?

• Better start at $F_1$, not $F_0=0$... – lhf Jun 13 '16 at 19:19

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

$$\sqrt[n]P_n\approx\frac{\phi^{(n+1)/2}}{\sqrt5}\approx\frac{F_n}{\sqrt{5\phi}}.$$

• – lhf Jun 13 '16 at 19:43
• @lhf: glad to know that I equaled Wolfram. – Yves Daoust Jun 13 '16 at 19:51
• Thanks very much for your answer! – Dingo13 Jun 14 '16 at 14:27