a) \(A=1\cdot2\cdot3-2\cdot3\cdot4+3\cdot4\cdot5+...+98\cdot99\cdot100\)
\(\Rightarrow4A=4\cdot\left(1\cdot2\cdot3+2\cdot3\cdot4+...+98\cdot99\cdot100\right)\)
\(\Rightarrow4A=1\cdot2\cdot3\cdot4+2\cdot3\cdot4\cdot4+3\cdot4\cdot5\cdot4+...+98\cdot99\cdot100\cdot4\)
\(\Rightarrow4A=1\cdot2\cdot3\cdot4+2\cdot3\cdot4\cdot\left(5-1\right)+3\cdot4\cdot5\cdot\left(6-2\right)+....+98\cdot99\cdot100\cdot\left(101-97\right)\)
\(\Rightarrow4A=1\cdot2\cdot3\cdot4+2\cdot3\cdot4\cdot5-1\cdot2\cdot3\cdot4+3\cdot4\cdot5\cdot6-....-97\cdot98\cdot99\cdot100\)
\(\Rightarrow4A=\left(1\cdot2\cdot3\cdot4-1\cdot2\cdot3\cdot4\right)+\left(2\cdot3\cdot4\cdot5-2\cdot3\cdot4\cdot5\right)+...+98\cdot99\cdot100\cdot101\)
\(\Rightarrow4A=0+0+0+...+98\cdot99\cdot100\cdot101\)
\(\Rightarrow4A=98\cdot99\cdot100\cdot101\)
\(\Rightarrow A=\dfrac{98\cdot99\cdot100\cdot101}{4}\)
b) \(B=1+3^2+3^3+3^4+...+3^{99}\)
\(\Rightarrow3B=3\cdot\left(1+3^2+3^3+...+3^{99}\right)\)
\(\Rightarrow3B=3+3^3+3^4+3^5+...+3^{99}+3^{100}\)
\(\Rightarrow3B-B=\left(3+3^3+3^4+...+3^{100}\right)-\left(1+3^2+3^3+...+3^{99}\right)\)
\(\Rightarrow2B=\left(3^3-3^3\right)+\left(3^4-3^4\right)+...+\left(3-1\right)+\left(3^{100}-3^2\right)\)
\(\Rightarrow2B=2+3^{100}-3^2\)
\(\Rightarrow B=\dfrac{2+3^{100}-3^2}{2}\)
c) \(C=\left(9^{2023}-9^{2022}\right):9^{2022}\)
\(C=\left(9^{2023}-9^{2022}\right)\cdot\dfrac{1}{9^{2022}}\)
\(C=\dfrac{9^{2023}}{9^{2022}}-\dfrac{9^{2022}}{9^{2022}}\)
\(C=9^1-1\)
\(C=8\)
