Question

Cho A = (1/2² - 1) . (1/3² - 1) . … . (1/100² - 1) So sánh A và -½ Giúp mik với… Pls 🙏🏿

Answer 1

\(A=\left(\dfrac{1}{2^2}-1\right)\left(\dfrac{1}{3^2}-1\right).....\left(\dfrac{1}{100^2}-1\right)\)

\(\Rightarrow A=\dfrac{-3}{4}.\left(-\dfrac{8}{9}\right).....\left(-\dfrac{9999}{10000}\right)\)

\(\Rightarrow A=-\left(\dfrac{1.3}{2.2}.\dfrac{2.4}{3.3}.....\dfrac{99.101}{100.100}\right)\)

\(\Rightarrow A=-\left(\dfrac{1.2.3....99}{2.3.....100}.\dfrac{3.4.5.....101}{2.3....100}\right)\)

\(\Rightarrow A=-\left(\dfrac{1}{100}.\dfrac{101}{2}\right)\)

\(\Rightarrow A=-\dfrac{101}{200}< -\dfrac{100}{200}=-\dfrac{1}{2}\)