How to evaluate $\lim_{n \to \infty}{\frac{a_1 \cdot a_2\cdot a_3 \ldots a_{n}}{a_{n+1}}}$, given $a_{n+1} = a_{n}^{2} -1,a_1 = \sqrt{5}$?

Question

Let $a_{n+1} = a_{n}^{2} -1,a_1 = \sqrt{5}.$ How would one evaluate $$\lim_{n \to \infty}{\frac{a_1 \cdot a_2\cdot a_3 \ldots a_{n}}{a_{n+1}}}?$$

Added: Someone else asked me this question today, but unfortunately I don't know how to start.

Answer 1

Offset by one, this is a known sequence. Without the offset, we have $$a_{n}=\left\lceil c^{2^n}\right\rceil$$ (for $n\ge 2$) where $c$ is some constant between 1 and 2.

The fraction in the limit is approximated by $$\frac{\sqrt{5}c^{2^1}c^{2^2}\cdots c^{2^n}}{c^{2^{n+1}}}=\frac{\sqrt{5}c^{2^{n+1}-2}}{c^{2^{n+1}}}=\frac{\sqrt{5}}{c^2}$$