Ta có A là tích của 99 số âm nên A < 0, do đó:
\(A=-\left[\left(1-\dfrac{1}{2^2}\right).\left(1-\dfrac{1}{3^2}\right).\left(1-\dfrac{1}{4^2}\right)...\left(1-\dfrac{1}{100^2}\right)\right]\)
\(\Leftrightarrow A=-\left[\left(1-\dfrac{1}{4}\right).\left(1-\dfrac{1}{9}\right).\left(1-\dfrac{1}{16}\right)...\left(1-\dfrac{1}{10000}\right)\right]\)
\(\Leftrightarrow A=-\left(\dfrac{3}{2^2}.\dfrac{8}{3^2}.\dfrac{15}{4^2}...\dfrac{9999}{100^2}\right)\)
\(\Leftrightarrow A=-\left(\dfrac{1.3}{2^2}.\dfrac{2.4}{3^2}.\dfrac{3.5}{4^2}...\dfrac{99.101}{100^2}\right)\)
\(\Leftrightarrow A=-\left(\dfrac{1.2.3...98.99}{2.3.4...99.100}.\dfrac{3.4.5...100.101}{2.3.4...99.100}\right)\)
\(\Leftrightarrow A=-\left(\dfrac{1}{100}.\dfrac{101}{2}\right)\Rightarrow A=-\dfrac{101}{200}\)
\(\Rightarrow A< \dfrac{-1}{2}\)
Vậy \(A< \dfrac{-1}{2}\)
