Sửa đề: \(1\cdot2\cdot3+2\cdot3\cdot4+3\cdot4\cdot5+\cdots+2020\cdot2021\cdot2022\)
Ta có: \(1\cdot2\cdot3+2\cdot3\cdot4+3\cdot4\cdot5+\cdots+2020\cdot2021\cdot2022\)
\(=2\left(2-1\right)\left(2+1\right)+3\left(3-1\right)\left(3+1\right)+\cdots+2021\left(2021-1\right)\left(2021+1\right)\)
\(=2\left(2^2-1\right)+3\left(3^2-3\right)+\cdots+2021\left(2021^2-1\right)\)
\(=\left(2^3+3^3+\cdots+2021^3\right)-\left(2+3+\cdots+2021\right)\)
\(=\left(1^3+2^3+\cdots+2021^3\right)-\left(1+2+3+\cdots+2021\right)\)
\(=\left(1+2+3+\cdots+2021\right)^2-\left(1+2+3+\cdots+2021\right)\)
\(=\left(2021\cdot\frac{2022}{2}\right)^2-\frac{2021\cdot2022}{2}\)
\(=\left(2021\cdot1011\right)^2-2021\cdot1011\)
