Question

[Lê Trí Cường]

rút gọn biểu thức:1²+2²+3²+............+2018²

Answer 1

[Phạm Tuấn Đạt]

Đặt \(D=1^2+2^2+3^2+...+2018^2\)

\(D=1\left(2-1\right)+2\left(3-1\right)+3\left(4-1\right)+...+2018\left(2019-1\right)\)

\(D=1.2-1+2.3-2+3.4-3+...+2018.2019-2018\)

\(D=\left(1.2+2.3+...+2018.2019\right)-\left(1+2+3+...+2018\right)\)

Đặt \(A=1.2+2.3+...+2018.2019\)

\(\Rightarrow3A=1.2.3+2.3.\left(4-1\right)+...+2018.2019\left(2020-2017\right)\)

\(\Rightarrow3A=2018.2019.2010\Rightarrow A=\frac{2018.2019.2020}{3}\)

Đặt \(B=1+2+3+...+2018\)

\(B=\frac{\left(2018+1\right)\left(2018-1+1\right)}{2}=\frac{2019.2018}{2}\)

\(\Rightarrow D=A+B=\frac{2018.2019.2020}{3}+\frac{2019.2018}{2}\)

\(\Rightarrow D=\frac{2018.2019.2020.2+2019.2018.3}{6}\)