\(A=-\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)...\left(1-\frac{1}{100^2}\right)\)
\(=-\left(\frac{2^2-1}{2^2}\right)\left(\frac{3^2-1}{3^2}\right)...\left(\frac{100^2-1}{100^2}\right)\)
\(=-\frac{1.3.2.4.3.6...99.101}{2.2.3.3.4.4...100.100}=-\frac{1.2.3...99}{2.3.4...100}.\frac{3.4.5...101}{2.3.4...100}\)
\(=-\frac{1}{100}.\frac{101}{2}=-\frac{101}{200}< -\frac{100}{200}< -\frac{1}{2}\)
#)Giải :
\(A=\left(\frac{1}{2^2}-1\right)\left(\frac{1}{3^2}-1\right)...\left(\frac{1}{100^2}-1\right)\)
\(A=-\frac{3}{2^2}.\left(-\frac{8}{3^2}\right).\left(-\frac{15}{4^2}\right).....\left(-\frac{9999}{100^2}\right)\)
\(A=-\left(\frac{3}{2^2}.\frac{8}{3^2}.\frac{15}{4^2}.....\frac{9999}{100^2}\right)\)
\(A=-\left(\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}.....\frac{99.101}{100.100}\right)\)
\(A=-\left(\frac{1.2.3.....99}{2.3.4.....100}.\frac{3.4.5.....101}{2.3.4.....100}\right)\)
\(A=-\frac{101}{200}< -\frac{1}{2}\)
\(\Rightarrow A< -\frac{1}{2}\)
