## badreferences Group Title If $\left\{a_n\right\}\mid n\geq 0,a_1=5,a_{n+1}=a_n^2-2,n\in\mathbb{R}$ find $\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_1a_2\cdots a_n}$. 2 years ago 2 years ago

1. Mr.Math Group Title

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

2. badreferences Group Title

$$\sqrt[2]{21}$$ actually.

3. Mr.Math Group Title

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

4. Mr.Math Group Title

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

5. badreferences Group Title

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.