2162.11 – An Infinite Continued Fraction

The key to this problem is to notice that this fraction contains a copy of itself, as indicated in the figure. Therefore,

$x = \frac{1}{1+\frac{1}{3+x}} = \frac{1}{\frac{3+x+1}{3+x}} = \frac{3+x}{4+x}$

Cross-multiplying and gathering terms give the quadratic equation:

$x^2 + 3x -3=0,$

whose solutions are $x = (-3 ± \sqrt{21})/2$. As $x$ is clearly positive, we go with the plus sign, and find that $x = (-3 + \sqrt{21})/2 \approx 0.79.$