Question

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

So sánh A với \(\frac{-1}{2}\)

Answer 1

\(A=\left(\frac{1}{2^2}-1\right).\left(\frac{1}{3^2}-1\right).\left(\frac{1}{4^2}-1\right)...\left(\frac{1}{100^2}-1\right)\)

\(A=\frac{-3}{2^2}.\frac{-8}{3^2}...\frac{-9999}{100^2}\)

\(A=-\left(\frac{3}{2^2}.\frac{8}{3^2}...\frac{9999}{100^2}\right)\)(vì có 99 thừa số, mỗi thừa số là âm nên kết quả là âm)

\(A=-\left(\frac{1.3}{2.2}.\frac{2.4}{3.3}...\frac{99.101}{100.100}\right)\)

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

\(A=-\left(\frac{1}{100}.\frac{101}{2}\right)=-\frac{101}{200}\)

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