Question

Tính A= (1-\(\dfrac{1}{2}\)).(1-\(\dfrac{1}{3}\)).(1-\(\dfrac{1}{5}\)).(1-\(\dfrac{1}{6}\)). ... .(1-\(\dfrac{1}{19}\)).(1-\(\dfrac{1}{20}\)). So sánh A với \(\dfrac{1}{21}\).

Answer 1

Ta có: \(A=\left(1-\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right)\left(1-\dfrac{1}{4}\right)\left(1-\dfrac{1}{5}\right)...\left(1-\dfrac{1}{19}\right)\left(1-\dfrac{1}{20}\right)\)

\(=\dfrac{1}{2}.\dfrac{2}{3}.\dfrac{3}{4}.\dfrac{4}{5}....\dfrac{18}{19}.\dfrac{19}{20}\)

\(=\dfrac{1.2.3.4....18.19}{2.3.4.5....19.20}\)

\(=\dfrac{1}{20}\) \(>\dfrac{1}{21}\)

\(\Rightarrow A>\dfrac{1}{21}\)

Vậy \(A>\dfrac{1}{21}.\)