\(A=1\cdot2+2\cdot3+3\cdot4+...+99\cdot100\)
\(3A=3\left(1\cdot2+2\cdot3+3\cdot4+...+99\cdot100\right)\)
\(3A=1\cdot2\cdot3+2\cdot3\cdot\left(4-1\right)+...+99\cdot100\cdot\left(101-98\right)\)
\(3A=1\cdot2\cdot3+2\cdot3\cdot4-1\cdot2\cdot3+...+99\cdot100\cdot101-98\cdot99\cdot100\)
\(3A=99\cdot100\cdot101\Rightarrow A=\dfrac{99\cdot100\cdot101}{3}=333300\)
theo đề bài ta có:
\(A=1.2+2.3+3.4+....+99.100\)
\(\Rightarrow3A=1.2.3+2.3.3+3.4.3+....+99.100.3\)
\(\Leftrightarrow3A=1.2.3+2.3.\left(4-1\right)+3.4\left(5-2\right)+...+99.100.\left(101-98\right)\)
\(\Leftrightarrow3A=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+99.100\)
\(.101-98.99.100\)
\(\Leftrightarrow3A=99.100.101\)
\(\Rightarrow A=99.100.101:3\)
\(\Rightarrow A=333300\)
