## Solution

Let

$$x = \sqrt{n + \sqrt{n + \sqrt{n + \cdots}}}$$

We have

\begin{aligned} x &= \sqrt{x + n} \\ \implies x^2 &= n + x \\ \implies x &= \frac{1 + \sqrt{4n+1}}{2} \text{ as x is positive} \end{aligned}

Therefore,

$$\sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}} = \frac{1 + \sqrt{5}}{2}$$