\(A=\dfrac{2^2}{1\cdot3}\cdot\dfrac{3^2}{2\cdot4}\cdot\dfrac{4^2}{3\cdot5}\cdot...\cdot\dfrac{999^2}{998\cdot1000}\\ =\dfrac{2^2\cdot3^2\cdot4^2\cdot...\cdot999^2}{1\cdot3\cdot2\cdot4\cdot3\cdot5\cdot...\cdot998\cdot1000}\\ =\dfrac{\left(2\cdot3\cdot4\cdot...\cdot999\right)\cdot\left(2\cdot3\cdot4\cdot...\cdot999\right)}{\left(1\cdot2\cdot3\cdot...\cdot998\right)\cdot\left(3\cdot4\cdot5\cdot...\cdot1000\right)}\\ =\dfrac{2\cdot3\cdot4\cdot...\cdot999}{1\cdot2\cdot3\cdot...\cdot998}\cdot\dfrac{2\cdot3\cdot4\cdot...\cdot999}{3\cdot4\cdot5\cdot...\cdot1000}\\ =999\cdot\dfrac{1}{500}\\ =\dfrac{999}{500}\)
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