Question

Cho A= (\(\dfrac{1}{2^2}\)-1).(\(\dfrac{1}{3^2}\)-1)...(\(\dfrac{1}{100^2}\)-1) So sánh A vs -\(\dfrac{1}{2}\)

Answer 1

\(A=\left(\dfrac{1}{2^2}-1\right)\left(\dfrac{1}{3^2}-1\right)....\left(\dfrac{1}{100^2}-1\right)\\ =-\dfrac{3}{4}.-\dfrac{8}{9}...-\dfrac{9999}{10000}\\ =-\left(\dfrac{3}{4}.\dfrac{8}{9}...\dfrac{9999}{10000}\right)\\ =-\left(\dfrac{1.3}{2.2}.\dfrac{2.4}{3.3}....\dfrac{99.101}{100.100}\right)=-\left(\dfrac{1.2....99}{2.3....100}.\dfrac{3.4....101}{2.3....100}\right)\\ =-\left(\dfrac{1}{100}.\dfrac{101}{2}\right)\\ =-\dfrac{101}{200}< \dfrac{-100}{200}=-\dfrac{1}{2}\\ \Rightarrow A< \dfrac{-1}{2}\)