14. [8] Find the exact value of $1+\frac{1}{1+\frac{2}{1+\frac{1}{1+\frac{2}{1+\ldots}}}}$.
Solution: Let $x$ be what we are trying to find. $x-1=\frac{1}{1+\frac{2}{1+\frac{1}{1+\frac{2}{1+\ldots}}}} \Rightarrow \frac{1}{x-1}-1=$ $\frac{2}{1+\frac{1}{1+\frac{2}{2}+}} \Rightarrow \frac{2}{\frac{1}{x-1}-1}=x \Rightarrow x^{2}-2=0$, so $x=\sqrt{2}$ since $x>0$.