# How can I write this as a recurrence relation?

$$\sqrt{5 + \sqrt{6 + \sqrt{7 + ... }}}$$

Thank you

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Let $a_n = \sqrt{ n+4 + \sqrt{n+5 + \sqrt{ n+ 6 + \sqrt{ \ldots } } } }$
Hint: Consider $a_n^2 - a_{n+1}$.
I've got $$a_{n+1} = a_{n}^{2} - (n+4)$$ but can this be simplified further? :S – user78416 May 28 '13 at 14:34
Sorry to bother you again, but I was hoping more for a recurrence relation to give the radical sequentially, like: $$x_{1} = \sqrt{5}, x_{2} = \sqrt{5 + \sqrt{6}}, x_{3} = \sqrt{5 + \sqrt{6 + \sqrt{7}}}$$ and so on. With this, if I define $$a_{1} = \sqrt{5}$$ then I get $$a_{2} = (\sqrt{5})^{2} - (2 + 4) = -1$$ Or have I done something stupid? – user78416 May 28 '13 at 14:54