Question

Chứng minh rằng: 1/5 + 1/15 + 1/25 +...+ 1/1985 < 9/20

Answer 1

Đặt \(A=\dfrac{1}{5}+\dfrac{1}{15}+\dfrac{1}{25}+...+\dfrac{1}{1985}\)

\(A=\dfrac{1}{5}.\left(1+\dfrac{1}{3}+\dfrac{1}{5}+...+\dfrac{1}{397}\right)\)

\(A=\dfrac{1}{5}.\left(1+\dfrac{1}{1+2}+\dfrac{1}{2+3}+...+\dfrac{1}{198+199}\right)\)

\(A=\dfrac{1}{5}.\left(1+1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{198}-\dfrac{1}{199}\right)\)

\(A=\dfrac{1}{5}.\left(2-\dfrac{1}{199}\right)\)

\(A=\dfrac{397}{995}< \dfrac{9}{20}\)

\(\Rightarrow\dfrac{1}{5}+\dfrac{1}{15}+\dfrac{1}{25}+...+\dfrac{1}{1985}< \dfrac{9}{20}\left(đpcm\right)\)