\(\left(\frac{1}{2^2}-1\right).\left(\frac{1}{3^2}-1\right).\left(\frac{1}{4^2}-1\right)...\left(\frac{1}{100^2}-1\right)\)
\(=\left(\frac{1}{4}-1\right).\left(\frac{1}{9}-1\right).\left(\frac{1}{16}-1\right)...\left(\frac{1}{10000}-1\right)\)
\(=-\frac{3}{4}.-\frac{8}{9}.\frac{-15}{16}...-\frac{9999}{10000}\)
\(=\left(-1\right)^{50}.\frac{3}{4}.\frac{8}{9}.\frac{15}{16}...\frac{9999}{10000}\)
\(=1.\frac{3}{4}.\frac{8}{9}.\frac{15}{16}...\frac{9999}{10000}\)
\(=\frac{3.8.15...9999}{4.9.16...10000}\)
\(=\frac{3.2.4.3.5...99.101}{2.2.3.3.4.4...100.100}\)
\(=\frac{\left(2.3.4...99\right).\left(3.4.5...101\right)}{\left(2.3...100\right).\left(2.3...100\right)}\)
\(=\frac{1.101}{100.2}\)
\(=\frac{101}{200}\)
Tham khảo nha !!!
