\(\left(1-\dfrac{1}{2}\right)\cdot\left(1-\dfrac{1}{3}\right)\cdot\left(1-\dfrac{1}{4}\right)\cdot\left(1-\dfrac{1}{5}\right)\cdot...\cdot\left(1-\dfrac{1}{2022}\right)\)
\(=\left(\dfrac{2}{2}-\dfrac{1}{2}\right)\cdot\left(\dfrac{3}{3}-\dfrac{1}{3}\right)\cdot\left(\dfrac{4}{4}-\dfrac{1}{4}\right)\cdot...\cdot\left(\dfrac{2022}{2022}-\dfrac{1}{2022}\right)\)
\(=\dfrac{1}{2}\cdot\dfrac{2}{3}\cdot\dfrac{3}{4}\cdot\dfrac{4}{5}\cdot\dfrac{5}{6}\cdot...\cdot\dfrac{2021}{2022}\)
\(=\dfrac{1\cdot2\cdot3\cdot4\cdot...\cdot2021}{2\cdot3\cdot4\cdot5\cdot6\cdot...\cdot2022}\)
\(=\dfrac{1}{2022}\)
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