Infinite...or finite?

Let $y = \sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}}$ . Then we have

$\begin{aligned} y^2 & = x + \sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}} \\ y^2 & = x + y \\ \implies x & = y^2 - y = y(y-1) \end{aligned}$

Since $0 \le x \le 200$ , we have $0 \le y(y-1) \le 200$ . Note that $y \ge 0$ and the possible integer $y$ is given by $0 \le y \le \left \lfloor \sqrt{200} \right \rfloor = 14$ . Since $y=0$ and $y=1$ both produce $x=0$ , there are only 14 possible integer values of $x \in [0, 200]$ for integer $y \in [1, 14]$ .

Then the sum of integer $x \in [0.200]$ is $\displaystyle S = \sum_{y=1}^{14} y(y-1) = \sum_{y=1}^{14} (y^2 - y) = \frac {14(15)(29)}6 - \frac {14(15)}2 = 910$ and divided by the number of values of $x$ , $\dfrac {910}{14} = \boxed{65}$ .