Example 1. (1983 Shanghai Mathematics Competition) For a natural number $n$, consider the quadratic equation $x^{2}+(2 n+1) x+n^{2}=0$, with its two roots being $\alpha_{\mathrm{n}}, \beta_{\mathrm{n}}$. Find the value of the following expression:
$$
\begin{array}{l} 
\frac{1}{\left(\alpha_{3}+1\right)\left(\beta_{3}+1\right)}+\frac{1}{\left(\alpha_{4}+1\right)} \frac{1}{\left(\beta_{4}+1\right)}+\cdots \\
+\frac{1}{\left(\alpha_{20}+1\right)\left(\beta_{20}+1\right)}
\end{array}
$$
\[
\begin{array}{l}
\text { Solve } \alpha_{\mathrm{n}}+\beta_{\mathrm{n}}=(2 n+1), \alpha_{\mathrm{n}} \beta_{\mathrm{n}}=n^{2}, \\
\text { then }\left(\alpha_{n}+1\right)\left(\beta_{n}+1\right)=n(n-2). \\
\text { The original expression }=\frac{1}{3 \times 1}+\frac{1}{4 \times 2}+\cdots+\frac{1}{20 \times 18} \\
=\frac{1}{2}\left[\left(1-\frac{1}{3}\right)+\left(\frac{1}{2}-\frac{1}{4}\right)+\right. \\
\left.\quad\left(\frac{1}{3}-\frac{1}{5}\right)+\cdots+\left(\frac{1}{18}-\frac{1}{20}\right)\right] \\
=\frac{531}{760}.
\end{array}
\]