Question

Tính giá trị của biểu thức:

S=\(\dfrac{1}{2022}\)- \(\dfrac{5}{2.4}\)-\(\dfrac{5}{4.6}\)-...-\(\dfrac{5}{2020.2022}\)

Answer 1

\(S=\dfrac{1}{2022}-\dfrac{5}{2\cdot4}-\dfrac{5}{4\cdot6}-...-\dfrac{5}{2020\cdot2022}\)

\(=\dfrac{1}{2022}-\dfrac{5}{2}\left(\dfrac{2}{2\cdot4}+\dfrac{2}{4\cdot6}+...+\dfrac{2}{2020\cdot2022}\right)\)

\(=\dfrac{1}{2022}-\dfrac{5}{2}\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+...+\dfrac{1}{2020}-\dfrac{1}{2022}\right)\)

\(=\dfrac{1}{2022}-\dfrac{5}{2}\left(\dfrac{1}{2}-\dfrac{1}{2022}\right)\)

\(=\dfrac{1}{2022}-\dfrac{5}{2}\cdot\dfrac{505}{1011}=\dfrac{1}{2022}-\dfrac{2525}{2022}\)

\(=\dfrac{-2524}{2022}=-\dfrac{1262}{1011}\)