A = \(\left(\frac{1}{2^2}-1\right)\left(\frac{1}{3^2}-1\right)\left(\frac{1}{4^2}-1\right)...\left(\frac{1}{100^2}-1\right)\)
= \(\frac{-3}{2^2}.\frac{-8}{3^2}.\frac{-15}{4^2}...\frac{-9999}{100^2}=-\frac{3.8.15...9999}{2.2.3.3.4.4...100.100}=-\frac{1.3.2.4.3.5...99.101}{2.2.3.3.4.4...100.100}\)
= \(-\frac{\left(1.2.3...99\right)\left(3.4.5...101\right)}{\left(2.3.4...100\right)\left(2.3.4...100\right)}=-\frac{1.101}{100.2}=\frac{-101}{200}< \frac{-100}{200}=-\frac{1}{2}\)
=> A < - 1/2
\(A=\left(\frac{1}{2}-1\right)\left(\frac{1}{2}+1\right)\left(\frac{1}{3}-1\right)\left(\frac{1}{3}+1\right)\left(\frac{1}{4}-1\right)\left(\frac{1}{4}+1\right)...\left(\frac{1}{100}-1\right)\left(\frac{1}{100}+1\right)\)
Xét \(B=\left(\frac{1}{2}-1\right)\left(\frac{1}{3}-1\right)\left(\frac{1}{4}-1\right)...\left(\frac{1}{100}-1\right)=\left(\frac{-1}{2}\right)\left(\frac{-2}{3}\right)\left(\frac{-3}{4}\right)...\left(\frac{-99}{100}\right)\)
Có 99 số hạng nhân với nhau nên kết quả cuối sẽ nhận dấu âm--->\(B=\frac{-1}{100}\)
Xét \(C=\left(\frac{1}{2}+1\right)\left(\frac{1}{3}+1\right)\left(\frac{1}{4}+1\right)...\left(\frac{1}{100}+1\right)=\frac{3}{2}.\frac{4}{3}.\frac{5}{4}...\frac{101}{100}=\frac{101}{2}\)
\(A=B.C=\frac{-1}{100}.\frac{101}{2}=\frac{-101}{200}< \frac{-100}{200}=\frac{-1}{2}\)