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## anonymous 4 years ago If $\left\{a_n\right\}\mid n\geq 0,a_1=5,a_{n+1}=a_n^2-2,n\in\mathbb{R}$ find $\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_1a_2\cdots a_n}$.

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1. Mr.Math

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

2. anonymous

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

3. Mr.Math

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

4. Mr.Math

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

5. anonymous

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.

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