Question

cho A=\(\dfrac{2}{3}\cdot\dfrac{4}{5}\cdot\dfrac{6}{7}\cdot...\cdot\dfrac{98}{99}\)

Chứng minh rằng A\(< \dfrac{1}{7}\)

Answer 1

Ta có:\(A=\dfrac{2}{3}\cdot\dfrac{4}{5}\cdot\dfrac{6}{7}\cdot...\cdot\dfrac{98}{99}\)

\(A< \dfrac{3}{4}\cdot\dfrac{5}{6}\cdot\dfrac{7}{8}\cdot...\cdot\dfrac{99}{100}\)

\(\Rightarrow A^2< \dfrac{2}{3}\cdot\dfrac{3}{4}\cdot\dfrac{4}{5}\cdot\dfrac{5}{6}\cdot\dfrac{6}{7}\cdot\dfrac{7}{8}\cdot...\cdot\dfrac{98}{99}\cdot\dfrac{99}{100}\)

\(A^2< \dfrac{2}{100}=\dfrac{1}{50}\)

Mà \(\dfrac{1}{50}< \dfrac{1}{49}\)

\(\Rightarrow A^2< \dfrac{1}{49}\)

\(\Rightarrow A< \dfrac{1}{7}\left(đpcm\right)\)