G4.4 Find the positive real solution of the equation $x=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{x}}}}$.
Determine the positive real root of the following equation: $x=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{x}}}}$.
$$
\begin{array}{l}
\frac{1}{x-1}=1+\frac{1}{1+\frac{1}{1+\frac{1}{x}}} \Rightarrow \frac{1}{x-1}-1=\frac{1}{1+\frac{x}{x+1}} \\
\frac{2-x}{x-1}=\frac{1}{1+\frac{x}{x+1}} \\
\frac{2-x}{x-1}=\frac{1}{\frac{2 x+1}{x+1}} \\
\frac{2-x}{x-1}=\frac{x+1}{2 x+1} \\
(2-x)(2 x+1)=(x-1)(x+1) \\
-2 x^{2}+3 x+2=x^{2}-1 \\
3 x^{2}-3 x-3=0 \\
x^{2}-x-1=0 \\
x=\frac{1+\sqrt{5}}{2} \text { or } \frac{1-\sqrt{5}}{2} \quad(<0, \text { rejected })
\end{array}
$$

Remark: “求賂数……的正数值。” is changed into “求下列方程……的正賂数解。”
"Determine the positive value of the real number ....." is changed into "Determine the positive real root of the following equation ......"