\(A=\dfrac{100}{1\cdot2}+\dfrac{100}{2\cdot3}+\dfrac{100}{3\cdot4}+...+\dfrac{100}{99\cdot100}\)
\(A=100\cdot\left(\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{99\cdot100}\right)\)
\(A=100\cdot\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{99}-\dfrac{1}{100}\right)\)
\(A=100\cdot\left(1-\dfrac{1}{100}\right)\)
\(A=100\cdot\dfrac{99}{100}\)
A=99
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