Question

a=1/1x3=1/3x5+1/5x7+....+1/2019x2021+1/2021x2023

Answer 1

Sửa đề: \(a=\frac{1}{1\cdot3}+\frac{1}{3\cdot5}+\cdots+\frac{1}{2019\cdot2021}+\frac{1}{2021\cdot2023}\)

Ta có: \(a=\frac{1}{1\cdot3}+\frac{1}{3\cdot5}+\cdots+\frac{1}{2019\cdot2021}+\frac{1}{2021\cdot2023}\)

\(=\frac12\left(\frac{2}{1\cdot3}+\frac{2}{3\cdot5}+\cdots+\frac{2}{2019\cdot2021}+\frac{2}{2021\cdot2023}\right)\)

\(=\frac12\left(1-\frac13+\frac13-\frac15+\cdots+\frac{1}{2019}-\frac{1}{2021}+\frac{1}{2021}-\frac{1}{2023}\right)\)

\(=\frac12\left(1-\frac{1}{2023}\right)=\frac12\cdot\frac{2022}{2023}=\frac{1011}{2023}\)