Question

If \[\left\{a_n\right\}\mid n\geq 0,a_1=5,a_{n+1}=a_n^2-2,n\in\mathbb{R}\] find \[\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_1a_2\cdots a_n}\].

Answer 1

This is just a guess: \(\large \frac{23}{5}\)?

Answer 2

\(\sqrt[2]{21}\) actually.

Answer 3

You didn't need to write the answer! -.-

Answer 4

Okay. I will try to write a proof for that.

Answer 5

Oh, haha. XD Apparently, the problem can be solved using repeated telescopy \(a+\frac{1}{a}=5\Rightarrow x_{n+1}={a^2}^n+\frac{1}{{a^2}^n}\), but I'm not sure I understand what they're getting at with this hint.