NL
1
K
Khách
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NH
1
NN
tinh: (1+\(\frac{1}{1.3}\))(1+\(\frac{1}{2.4}\))(1+\(\frac{1}{3.5}\)).......(1+\(\frac{1}{99.101}\))
1
19 tháng 8 2018
Ta có: \(\left(1+\frac{1}{1.3}\right)\left(1+\frac{1}{2.4}\right)\left(1+\frac{1}{3.5}\right)...\left(1+\frac{1}{99.101}\right)\)
\(=\left(1+\frac{1}{3}\right)\left(1+\frac{1}{8}\right)\left(1+\frac{1}{15}\right)...\left(1+\frac{1}{9999}\right)\)
\(=\frac{4}{3}.\frac{9}{8}.\frac{16}{15}.....\frac{10000}{9999}\)
\(=\frac{2.2}{1.3}.\frac{3.3}{2.4}.\frac{4.4}{3.5}.....\frac{100.100}{99.101}\)
\(=\frac{2.2.3.3.4.4.....100.100}{1.3.2.4.3.5.....99.101}\)
\(=\frac{\left(2.3.4.....100\right)\left(2.3.4.....100\right)}{\left(1.2.3.....99\right)\left(3.4.5.....101\right)}\)
\(=\frac{2.3.4.....100}{1.2.3.....99}.\frac{2.3.4.....100}{3.4.5.....101}\)
\(=100.\frac{2}{101}\)
\(=\frac{200}{101}\)
TP
8 tháng 8 2017
Ta có:
\(\frac{1}{1.3}+\frac{1}{2.4}+\frac{1}{3.5}+\frac{1}{4.6}+\frac{1}{5.7}+\frac{1}{6.8}+\frac{1}{7.9}+\frac{1}{8.10}\)
\(=\frac{1}{2}.\left(\frac{1}{1}-\frac{1}{3}+\frac{1}{2}-\frac{1}{4}+\frac{1}{3}-\frac{1}{5}+....+\frac{1}{8}-\frac{1}{10}\right)\)
\(=\frac{1}{2}.\left(\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}....+\frac{1}{7}-\frac{1}{9}\right)+\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+...+\frac{1}{8}-\frac{1}{10}\right)\)
\(=\frac{1}{2}.\frac{8}{9}+\frac{1}{2}.\frac{2}{5}=\frac{1}{2}.\left(\frac{8}{9}+\frac{2}{5}\right)=\frac{1}{2}.\frac{58}{45}=\frac{29}{45}\)
Ta có :\(\left(1+\frac{1}{1.3}\right)+\left(1+\frac{1}{2.4}\right)+...+\left(1+\frac{1}{18.20}\right)\)
= \(\frac{4}{1.3}+\frac{9}{2.4}+...+\frac{361}{18.20}\)
= \(\frac{2.2.3.3.4.4.....18.18.19.19}{1.3.2.4.3.5.....17.19.18.20}\)
= \(\frac{2.19}{1.20}=\frac{19}{10}\)