<s>[INST] Limit of $y=\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-\sqrt{5+\ldots}}}}}$ I would appreciate any help with this problem:
If 
$$y=\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-\sqrt{5+\ldots}}}}}$$
Then how do I find $y^2 - y$?
I'm not sure whether this is an arithmetic or geometric series. 
 [/INST] Note that $(y^2-5)^2=5-y$. It is also clear that $ y^2 \geq 5$ and  $0<y$.
We have $0=y^4-10y^2+y+20=(y^2-y-4)(y^2+y-5)$.
We have $y^2+y-5>0$, so $y^2-y=4$.
P.S. Incidentally, if you want to find $y$, it is the positive root of $y^2-y-4=0$, which is $\frac{1+\sqrt{17}}{2}$.
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