\(A=1\cdot2\cdot3+2\cdot3\cdot4+...+1998\cdot1999\cdot2000\)
\(\Rightarrow4A=1\cdot2\cdot3\cdot4+2\cdot3\cdot4\cdot4+....+1998\cdot1999\cdot2000\cdot4\)
\(\Rightarrow4A=1\cdot2\cdot3\cdot\left(4-0\right)+2\cdot3\cdot4\left(5-1\right)+....+1998\cdot1999\cdot2000\left(2001-1997\right)\)
\(\Rightarrow4A=1\cdot2\cdot3\cdot4-0\cdot1\cdot2\cdot3+2\cdot3\cdot4\cdot5-1\cdot2\cdot3\cdot4+...+1998\cdot1999\cdot2000\cdot2001-1997\cdot1998\cdot1999\cdot2000\)\(\Rightarrow4A=1997\cdot1998\cdot1999\cdot2000\)
\(\Rightarrow A=\frac{1997\cdot1998\cdot1999\cdot2000}{4}\)
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