Example 1. Find $\mathrm{S}_{\mathrm{n}}=1 \cdot 2 \cdot 4+2 \cdot 3 \cdot 5+\cdots$ $+n(n+1)(n+3)$.
\begin{array}{l}\text { Solve } u_{k}=k(k+1)(k+3) \\ =k(k+1)(k+2+1)=k(k+1)(k+2) \\ +k(k+1), \text { which meets the condition of using ( } *) \text { formula. } \\ \text { First, find } \\ S_{n}^{1}=\sum_{k=1}^{n} k(k+1)(k+2), \text { where } \\ a=0, b=1, r=3. \\ \text { Then, } S_{n}^{1}=\frac{(n+3)n(n+1)(n+2)}{4 \cdot 1}+c. \\ \text { And } S_{1}^{1}=1 \cdot 2 \cdot 3=\frac{4 \cdot 1 \cdot 2 \cdot 3}{4}+c, \quad \therefore c=0. \\ \therefore S_{n}^{1}=\frac{n(n+1)(n+2)(n+3)}{4}, \\ S_{n}^{2}=\sum_{k=1}^{n} k(k+1)=\frac{1}{3} n(n+1)(n+2), \\ \text { Therefore, } S_{n}=\sum_{k=1}^{n} u_{k}=S_{n}^{1}+S_{n}^{2} \\ =\frac{n(n+1)(n+2)(3 n+13)}{12} \\\end{array}