Đặt tổng trên là A
\(\dfrac{2A}{7}=\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+...+\dfrac{2}{2021.2023}=\)
\(=\dfrac{3-1}{1.3}+\dfrac{5-3}{3.5}+\dfrac{7-5}{5.7}+...+\dfrac{2023-2021}{2021.2023}=\)
\(=1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{2021}-\dfrac{1}{2023}=\)
\(=\dfrac{2022}{2023}\Rightarrow A=\dfrac{7.2022}{2.2023}\)
\(A=\dfrac{7}{1.3}+\dfrac{7}{3.5}+\dfrac{7}{5.7}+...+\dfrac{7}{2021.2023}\\ \Rightarrow\dfrac{2}{7}A=\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+...+\dfrac{2}{2021.2023}\\ \Rightarrow\dfrac{2}{7}A=2-\dfrac{2}{3}+\dfrac{2}{3}-\dfrac{2}{5}+\dfrac{2}{5}-\dfrac{2}{7}+...+\dfrac{2}{2021}-\dfrac{2}{2023}\\ \Rightarrow\dfrac{2}{7}A=2-\dfrac{2}{2023}=\dfrac{4044}{2023}\Rightarrow A=\dfrac{2022}{289}\)
A = 7/2 . (2/1.3 + 2/3.5 + ... + 2/2021.2023)
= 7/2 . (1 - 1/3 + 1/3 - 1/5 + .... + 1/2021 - 1/2023)
= 7/2 . (1- 1/2023)
= 7/2 . 2022/2023
=7.2.1011/ (2.2023)
=7077/2023
