anonymous
  • anonymous
If \[\left\{a_n\right\}\mid n\geq 0,a_1=5,a_{n+1}=a_n^2-2,n\in\mathbb{R}\] find \[\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_1a_2\cdots a_n}\].
Mathematics
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anonymous
  • anonymous
If \[\left\{a_n\right\}\mid n\geq 0,a_1=5,a_{n+1}=a_n^2-2,n\in\mathbb{R}\] find \[\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_1a_2\cdots a_n}\].
Mathematics
katieb
  • katieb
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Mr.Math
  • Mr.Math
This is just a guess: \(\large \frac{23}{5}\)?
anonymous
  • anonymous
\(\sqrt[2]{21}\) actually.
Mr.Math
  • Mr.Math
You didn't need to write the answer! -.-

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Mr.Math
  • Mr.Math
Okay. I will try to write a proof for that.
anonymous
  • 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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