$$
\text { 4.62 Solve the equation } \frac{x}{2+\frac{x}{2+\frac{x}{2+\ddots \frac{x}{2+\frac{x}{1+\sqrt{1+x}}}}}}=1
$$
(In the above expression, there are 1985 twos.)
[Solution] Since $x=0$ is not a solution to the original equation, we may assume $x \neq 0$, and thus $\sqrt{1+x} - 1 \neq 0$. Therefore, we have
$$
2+\frac{x}{1+\sqrt{1+x}}=2+(\sqrt{1+x}-1)=1+\sqrt{1+x} .
$$

The above equation shows that the original equation can be transformed into
$$
\frac{x}{1+\sqrt{1+x}}=1,
$$

Rationalizing the denominator, we get $\sqrt{1+x}-1=1$,
thus $\quad x=3$.
Upon verification, $x=3$ is a solution to the original equation.