\(=-\left(\frac{3}{4}.\frac{8}{9}.\frac{15}{16}...\frac{9800}{9801}\right)\)
\(=-\left(\frac{1.3}{2.2}.\frac{2.4}{3.3}...\frac{98.100}{99.99}\right)\)
\(=-\left(\frac{1.2.3....98}{2.3...99}.\frac{3.4.5...100}{2.3...99}\right)\)
\(=-\left(\frac{1}{99}.50\right)\)
\(=-\frac{50}{99}\)
Chúc hok tốt!!!
\(\frac{3}{4}.\frac{8}{9}.\frac{15}{16}...\left(\frac{1}{99^2}-1\right)\)
= \(\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}...\frac{98.100}{99.99}\)
= \(\frac{\left(1.2.3...98\right).\left(3.4.5...100\right)}{\left(2.3.4...99\right).\left(2.3.4...99\right)}\)
= \(\frac{100}{99.2}\)= \(\frac{50}{99}\)
\(\left(\frac{1}{2^2}-1\right).\left(\frac{1}{3^2}-1\right).\left(\frac{1}{4^2}-1\right).....\left(\frac{1}{99^2}-1\right)\)
\(=\left(-\frac{3}{4}\right).\left(-\frac{8}{9}\right).\left(-\frac{15}{16}\right).....\left(-\frac{99}{100}\right)\)
Ta thấy dãy trên có 98 phân số nên tích của các phân số trên là số dương
\(=\frac{3}{4}.\frac{8}{9}.\frac{15}{16}....\frac{99}{100}\)
\(=\frac{3.8.15...99}{4.9.16...100}=\frac{1.3.2.4.3.6...9.11}{2.2.3.3.4.4....10.10}=\frac{\left(1.2.3..9\right).\left(3.4.5...11\right)}{\left(2.3.4...10\right).\left(2.3.4....10\right)}\)
\(=\frac{1.11}{10.2}=\frac{11}{20}\)
