Question

\(\frac{1}{10.9}+\frac{1}{18.13}+\frac{1}{26.17}+...+\frac{1}{802.405}\)

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Answer 1

Đặt Biểu thức trên là A

\(A=\frac{1}{2.5.9}+\frac{1}{2.9.13}+\frac{1}{2.13.17}+...+\frac{1}{2.397.401}+\frac{1}{2.401.405}\)

\(A=\frac{1}{2}\left(\frac{1}{5.9}+\frac{1}{9.13}+\frac{1}{13.17}+...+\frac{1}{397.401}+\frac{1}{401.405}\right)\)

\(4A=\frac{1}{2}\left(\frac{4}{5.9}+\frac{4}{9.13}+\frac{4}{13.17}+...+\frac{4}{397.401}+\frac{4}{401.405}\right)\)

\(4A=\frac{1}{2}\left(\frac{9-5}{5.9}+\frac{13-9}{9.13}+\frac{17-13}{13.17}+...+\frac{401-397}{397.401}+\frac{405-401}{401.405}\right)\)

\(4A=\frac{1}{2}\left(\frac{1}{5}-\frac{1}{405}\right)=\frac{1}{2}.\frac{80}{405}=\frac{40}{405}\Rightarrow A=\frac{40}{4.405}=\frac{2}{81}\)