consider the equation

$\displaystyle x=\sqrt{x+2\sqrt{x+2\sqrt{x+...+2\sqrt{x+2\sqrt{x+ 2x}}}}},$

with n square roots.

I'm asked to show that the real roots of this equation are independent of n, and to find them.

Now i know that the real roots are x=0 and x=3, however my problem lies in showing that these are independent of n. I considered showing it by induction, but my induction relies on already assuming that the solutions are independent of n, and in my opinion that would be invalid as a proof. I vaguely remember doing some work on continued fractions, and was wondering if it is possible to represent the above equation as an infinite continued fraction, which would then show that the solution is independent of n, but I can't seem to find an infinite continued fraction for this expression. Any ideas?

Thanks in advance.