Example 9. Let $x=\frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n+1}+\sqrt{n}}$, $y=\frac{\sqrt{n+1}+\sqrt{n}}{\sqrt{n+1}-\sqrt{n}}(n$ is a natural number).
Then when $n=$ $\qquad$, the value of the algebraic expression $19 x^{2}+123 x y+$ $19 y^{2}$ is 1985.
From the given information, we have $x+y=4n+2, xy=1$. Therefore,
$$
\begin{array}{l}
19x^2 + 123xy + 19y^2 \\
= 19(x+y)^2 - 38xy + 23xy \\
= 19(4n+2)^2 + 85.
\end{array}
$$

According to the problem, $19(4n+2)^2 + 85 = 1985$.