Example 20. Find the sum: $1^{2}+2^{2}+3^{2}+\cdots+n^{2}$.
\begin{aligned} \text { Sol } & \because n^{2}=2 \cdot \frac{n(n-1)}{2}+n \\ & =2 C_{2}^{2}+n=2\left(C_{n+1}^{3}-C_{n}^{3}\right)+n, \\ \therefore & \text { Original expression }=1^{3}+2^{2}+2\left[\left(C_{4}^{3}-C_{3}^{3}\right)\right. \\ + & \left.\left(C_{5}^{3}-C_{4}^{3}\right)+\cdots+\left(C_{n+1}^{3}-C_{n}^{3}\right)\right] \\ + & (3+4+5+\cdots+n) \\ = & 5+2\left(C_{n+1}^{3}-1\right)+\frac{(3+n)(n-2)}{2} \\ = & \frac{2(n+1) n(n-1)}{6}+\frac{n(n+1)}{2} \\ = & \frac{n(n+1)(2 n+1)}{6} .\end{aligned}