\(=\dfrac{1-2^2}{2^2}.\dfrac{1-3^2}{3^2}...\dfrac{1-100^2}{100^2}\)
\(=-\dfrac{2^2-1}{2^2}.\dfrac{3^2-1}{3^2}...\dfrac{100^2-1}{100^2}\)
\(=-\dfrac{(2-1)(2+1)}{2^2}.\dfrac{(3-1)(3+1)}{3^2}...\dfrac{(100-1)(100+1)}{100^2}\)
\(=-\dfrac{1.3}{2.2}.\dfrac{2.4}{3.3}.\dfrac{99.101}{100.100}\)
\(=-\dfrac{1.2.3...99}{2.3.4...100}.\dfrac{3.4...101}{2.3.4...100}\)
\(=-\dfrac{1}{100}.\dfrac{101}{2}\)
\(=>A=-\dfrac{101}{100.2}<-\dfrac{1}{2}(đpcm)\)