Does equation $a_n=\sqrt{a_{n-1}+6}$ with $a_1=6$ have a closed form? I've found no linearization method. Any suggestion or hint will be highly appreciated.
Closed form for non-linear recurrence $a_n=\sqrt{a_{n-1}+6}$ with $a_1=6$
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recurrence-relations
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1It [seems](http://tinyurl.com/cw2p36y) to converge to $3$... – 2012-08-03
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1@draks: It does converge to $3$ (roughly like a geometric series of ratio $1/6$). – 2012-08-03
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1I would guess not, but it is going to be close to $3 + \frac{k_1}{6^n} + \frac{k_2}{6^{2n}} + \cdots$ for some $k_1,k_2,\ldots$ (in this case it seems $k_1 \approx 16.4558$ and $k_2 \approx 0.614$). – 2012-08-03
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0If the question is $a_n=\sqrt{a_{n-1}+2}$ instead I can solve this explicitally. – 2012-08-04
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0@doraemonpaul Then you should post this here (and probably mention that your solution is direct when $|a_1|\leqslant2$ but that it needs some adjustment otherwise). – 2012-09-08