Problem - 2153
Find the number $x = [1, 2, 3, 1, 2, 3, \cdots]$. (continued fraction)
We have $$x = 1 + \frac{1}{2+ \displaystyle\frac{1}{3 + \displaystyle\frac{1}{x}}} = \displaystyle\frac{10x + 3}{7x + 2} \implies x=\boxed{\frac{4+\sqrt{37}}{7}}$$