Question

Tính giá trị của biểu thức sau:

A = \(\frac{1}{19}+\frac{9}{19\cdot29}+\frac{9}{29\cdot39}+......+\frac{9}{1999\cdot2009}\)

Answer 1

Ta có: \(A=\frac{1}{19}+\frac{9}{19.29}+\frac{9}{29.39}+...+\frac{9}{1999.2009}\)

\(\Rightarrow A=\frac{1}{19}+\frac{9}{10}\left(\frac{10}{19.29}+\frac{10}{29.39}+...+\frac{10}{1999.2009}\right)\)

\(\Rightarrow A=\frac{1}{19}+\frac{9}{10}\left(\frac{1}{19}-\frac{1}{29}+\frac{1}{29}-\frac{1}{39}+...+\frac{1}{1999}-\frac{1}{2009}\right)\)

\(\Rightarrow A=\frac{1}{19}+\frac{9}{10}\left(\frac{1}{19}-\frac{1}{2009}\right)\)

\(\Rightarrow A=\frac{1}{19}+\frac{9}{10}.\frac{1990}{38171}\)

\(\Rightarrow A=\frac{1}{19}+\frac{1791}{38171}\)

\(\Rightarrow A=\frac{200}{2009}\)

Vậy \(A=\frac{200}{2009}.\)