Question

[FFPUBGAOVCFLOL]

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

Hãy so sánh A với \(-\frac{1}{2}\)

Answer 1

[Inequalities]

\(A=\frac{-1.3}{2^2}.\frac{-2.4}{3^2}...\frac{-99.101}{100^2}\)

\(=-\left(\frac{1.2...99}{2.3...100}.\frac{3.4...101}{2.3...100}\right)\)

\(=-\left(\frac{1}{100}.\frac{101}{2}\right)\)

\(=-\frac{101}{200}< \frac{-100}{200}=\frac{-1}{2}\)