# A recurrence relation

by asmani
Tags: recurrence, relation
 P: 84 Hi all Suppose that $a_1=\sqrt5$, $a_{n+1}=a_n^2-2$ and $g_n=\frac{a_1a_2...a_n}{a_{n+1}}$. Evaluate $\lim_{n\rightarrow \infty } g_n$. I have seen some information in this link. Besides, the sequence gn seems as a good rational approximation for $\sqrt5$. I know that the answer is 1, But I can't find any nice solution. Any hint is strongly appreciated.
 HW Helper P: 3,515 I have yet to study recurrence relations for myself, but I played around with it for a bit and it seems like it was headed in the right direction. $$g_n=\frac{a_1a_2...a_n}{a_n^2-2}$$ $$=\frac{a_n^2\left(\frac{a_1a_2...a_{n-1}}{a_n}\right)-2\left(\frac{a_1a_2...a_{n-1}}{a_n}\right)+2\left(\frac{a_1a_2...a_{n-1}}{a_n}\right)}{a_n^2-2}$$ $$=\frac{\left(a_n^2-2\right)\left(\frac{a_1a_2...a_{n-1}}{a_n}\right)+2\left(\frac{a_1a_2...a_{n-1}}{a_n}\right)}{a_n^2-2}$$ $$=\frac{a_1a_2...a_{n-1}}{a_n}+2\frac{a_1a_2...a_{n-1}}{a_na_{n+1}}$$ $$=g_{n-1}+\frac{2}{a_n^2}g_n$$ Hence we can express gn in terms of gn-1, and the factor multiplying gn-1 seems to say a lot about what gn is as n gets very large.
 P: 84 Thanks. I can't see what does it say about gn. Can you help me on this? I found out how to show that: $$a_n=\frac{\varphi ^{2^n}-\varphi ^{-2^n}}{\varphi ^{2^{n-1}}-\varphi ^{-2^{n-1}}}$$ And then the rest is easy. But I'm still looking for a more elegant solution.
 HW Helper P: 3,515 A recurrence relation Hmm, after looking at it a bit more I realize it's really not telling us as much as I originally thought. After solving for gn in $$g_n=g_{n-1}+\frac{2}{a^2_n-2}g_n$$ to obtain $$g_n=\frac{a_n^2}{a_n^2-2}g_{n-1}$$ I was simply arguing that the factor approaches 1 as n gets large, but this doesn't really tell us about the limit of gn...
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Thanks
P: 26,157
hi asmani!
 Quote by asmani I found out how to show that: $$a_n=\frac{\varphi ^{2^n}-\varphi ^{-2^n}}{\varphi ^{2^{n-1}}-\varphi ^{-2^{n-1}}}$$ And then the rest is easy. But I'm still looking for a more elegant solution.
reverse-engineering that expression, we get
an = sinh(C2n)/sinh(C2n-1) = 2cosh(C2n-1)
sooo … try defining an = 2coshbn
 P: 84 Thanks. I think it's better not to simplify the fraction to get: $$g_n=a_1a_2\cdots a_n\frac{1}{a_{n+1}}=\frac{\sinh(C2^1) }{\sinh(C2^0)}\frac{\sinh(C2^2) }{\sinh(C2^1)}\cdots \frac{\sinh(C2^n) }{\sinh(C2^{n-1})}\frac{\sinh(C2^n) }{\sinh(C2^{n+1})}$$ $$=\frac{\sinh^2(C2^n) }{\sinh(C2^0)\sinh(C2^{n+1})}$$ Where C=Log(φ). And now calculating the limit is easy.
 HW Helper P: 3,515 Sorry about wasting your time asmani, I should've left it to the big guys