A problem on continued fractions Find the value of $x$, if:
$$\large 1+\frac{1}{2+\frac{1}{1+\frac{1}{2+...}}}$$
My attempt:
Noting that:
$$\large x=1+\frac{1}{2+\frac{1}{x}}$$
$$x=\frac{1+\sqrt{3}}{2}$$
question: Is my solution correct?

If others would like to know how he arrived at that:
Like you stated:
$$\large x=1+\frac{1}{2+\frac{1}{x}}$$
$$\large x= 1 + \frac{1}{\frac{2x+1}{x}}$$
$$\large x = 1 + \frac{x}{2x+1}$$
$$\large 2x^2 + x = 2x + 1 + x$$
$$\large 2x^2 - 2x - 1 =0$$
Quadratic Formula:
$$x = \frac{2 \pm \sqrt{4 + 8}}{4}$$
$$x = \frac{2 \pm 2\sqrt{3}}{4}$$
$$x = \frac{1 \pm \sqrt{3}}{2}$$
Because $x = \frac{1 - \sqrt{3}}{2}$ is extraneous, our final solution is
$$x = \frac{1 + \sqrt{3}}{2}$$
