Example 18. Find the sum: $1 \times 2+2 \times 3+3 \times 4$ $+\cdots+n(n+1)$
\[
\begin{array}{l}
\text { Solution } \because n(n+1)=2 \times \frac{n(n+1)}{1 \cdot 2} \\
=2 C_{n+1}^{2}=2\left[C_{n+2}^{2}-C_{n+1}^{3}\right], \\
\begin{aligned}
\therefore \text { Original expression } & =2\left[C_{2}^{2}+C_{3}^{2}+C_{1}^{2}+\cdots+C_{1+1}^{2}\right] \\
= & 2\left[1+C_{4}^{3}-C_{3}^{3}+C_{5}^{3}-C_{4}^{3}+\cdots\right. \\
& \left.+C_{n+2}^{3}-C_{n+1}^{3}\right] \\
& =2 C_{n+2}^{3}=\frac{n(n+1)(n+2)}{3} .
\end{aligned}
\end{array}
\]