$$\begin{gathered} (\mathrm{n}-1)\mathrm{n}(\mathrm{n}+1)=({\mathrm{n}}^2-\mathrm{n})(\mathrm{n}+1)=({\mathrm{n}}^2-\mathrm{n})\mathrm{n}+{\mathrm{n}}^2-\mathrm{n}={\mathrm{n}}^3-{\mathrm{n}}^2+{\mathrm{n}}^2-\mathrm{n}={\mathrm{n}}^3-\mathrm{n} \\ \Rightarrow {\mathrm{n}}^3=(\mathrm{n}-1)\mathrm{n}(\mathrm{n}+1) \end{gathered}$$

Bài ra:

$$\begin{gathered} \mathrm{A}=1\times 2^2+2\times 3^2+3\times 4^2+\mathrm{\dots }+99\times {100}^2 \\ =\left(2-1\right)\times 2^2+\left(3-1\right)\times 3^2+\left(4-1\right)\times 4^2+\mathrm{\dots }+\left(100-1\right)\times {100}^2 \\ =2^3+3^3+4^3+...+{100}^3-2^2-3^2-4^2-...-{100}^2 \\ =2^3+3^3+4^3+...+{100}^3-\left(2^2+3^2+4^2+...+{100}^2\right) \\ \mathrm{Đặt}: \\ \mathrm{B}=2^3+3^3+4^3+...+{100}^3 \\ \mathrm{C}=2^2+3^2+4^2+...+{100}^2 \\ \mathrm{Ta} \mathrm{có}: \\ \mathrm{B}=2^3+3^3+4^3+...+{100}^3 \\ =(2-1).2.(2+1)+2+(3-1).3.(3+1)+3+...+(100-1).100.(100+1)+100 \\ =2+1.2.3+3+2.3.4+...+100+99.100.101 \\ =(2+3+4+...+100)+(1.2.3+2.3.4+...+99.100.101) \\ =5049+25497450 \\ =25502499 \end{gathered}$$

$$\begin{gathered} \mathrm{C}=2^2+3^2+4^2+...+{100}^2 \\ = 2.2 + 3.3 +4.4...+ 100.100 \\ = 2. (3-1) + 3. (4-1)44+4.(5-1) +... + 100.(101 -1) \\ = 2.3 - 2 + 3.4 - 3 +4.5 - 4 +...+ 100. 101 - 100 \\ = ( 2.3 + 3.4 +4.5 +... + 100. 101) - ( 2 + 3+4+... + 100) \\ *) \mathrm{Xét}: 2.3 + 3.4 +4.5 +... + 100. 101 \\ \mathrm{Đặt} \mathrm{D} =2.3 + 3.4 +4.5 +... + 100. 101 \\ \Rightarrow 3\mathrm{D} = 3.(2.3 + 3.4 +4.5 +... + 100. 101) \\ = 2.3.3 + 3.4.3+3.4.5... + 100.101.3 \\ = 2.3. (4-1) + 3.4.(5-2) +...+ 100.101.(102-99) \\ = 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 +... + 100.101.102 - 99.100.101 \\ = 100.101.102 - 1.2.3 \\ \Rightarrow \mathrm{D}= 343398 \\ *) \mathrm{Xét}: 2 + 3+4+... + 100 \\ \mathrm{Đặt} \mathrm{E}=2 + 3+4+... + 100 \\ = 5049 \\ \Rightarrow \mathrm{C} = \mathrm{D} + \mathrm{E} = 343498 + 5049 = 348447 \\ \Rightarrow \mathrm{A}=\mathrm{B}-\mathrm{C}=25502499 -348447=25154052 \end{gathered}$$