Đặt A = 1.2 + 2.3 + 3.4 + ... + 2019.2020 + 2020.2021
=> 3A = 1.2.3 + 2.3.3 + 3.4.3 + ... + 2019.2020.3 + 2020.2021.3
=> 3A = 1.2.3 + 2.3.(4 - 1) + 3.4.(5 - 2) + ... + 2019.2020.(2021 - 2018) + 2020.2021.(2022 - 2019)
=> 3A = 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + ... + 2019.2020.2021 - 2018.2019.2020 + 2020.2021.2022 - 2019.2020.2021
=> 3A = 2020.2021.2022
=> A = 2 751 551 080
Đặt \(A=1.2+2.3+3.4+.........+2019.2020+2020.2021\)
\(\Rightarrow3A=1.2.3+2.3.3+3.4.3+.....+2019.2020.3+2020.2021.3\)
\(=1.2.3+2.3.\left(4-1\right)+3.4.\left(5-2\right)+.....+2020.2021.\left(2022-2019\right)\)
\(=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+2020.2021.2022-2019.2020.2021\)
\(=2020.2021.2022\)
\(\Rightarrow A=\frac{2020.2021.2022}{3}\)
