Hey,

I posted this before and didn't get any replies, so I'm posting again in case anyone can help. I've going round the houses with this problem and getting nowhere

. The problem is as follows

Consider the equation

x=\sqrt{x+2\sqrt{x+2\sqrt{x+...+2\sqrt{x+2x}}}}

where there are n square roots. Show that the real roots of this expression are independent of n and find them.

The real roots are x=3 and x=0. My proof for the fact that these are independent of n goes:

If we define F_1=x^2 and F_n=\left(\frac{F_{n-1}-x}{2}\right)^2 then we show by induction that F_n=3x has real roots x=3 and x=0 for all n\geqslant 1,\,n\in\mathbb{N}.

Clearly $F_1=3x\Rightarrow x^2=3x\Rightarrow x(x-3)=0$ has these real roots, so assuming this holds for n-1 the case for

is

F_n=3x\Rightarrow \left(\frac{F_{n-1}-x}{2}\right)^2=3x\Rightarrow \left(\frac{3x-x}{2}\right)^2=3x because F_n=3x for all n. This reduces again down to x(x-3)=0, so the real roots are independent of n.

My problem is that in all cases, F_{n-1} = x(x-3)p(x) for some polynomial p which has no real roots, and I don't think my proof as it stands is valid.

I'm desperate for some help with this problem it's been driving me mad, please help.

Thanks.