Finding the number represented by a continued fraction

Question

Suppose I have the following continued fraction expansion:

$[2,\overline{6,1}]=\alpha$

How do I find $\alpha$?

Answer 1

You have $\alpha=2+\frac{1}{\beta}$, where $\beta=[\overline{6,1}] $ is a number $>6$ fulfilling $$ \beta = 6+\frac{1}{1+\frac{1}{\beta}}=\frac{7\beta+6}{\beta+1}. $$ It follows that $\beta$ is a root of $z^2-6z-6$, hence $\beta=3+\sqrt{15}$ and $$ \alpha= 2+\frac{1}{\sqrt{15}+3} = 2+\frac{\sqrt{15}-3}{6}=\color{red}{\frac{9+\sqrt{15}}{6}}.$$