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BH
5 tháng 6 2016
Đặt \(A=1+\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{5050}\)
\(\Rightarrow\frac{1}{2}A=\frac{1}{2}\left(1+\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{5050}\right)\)
\(\Rightarrow\frac{1}{2}A=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{10100}\)
\(\Rightarrow\frac{1}{2}A=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{100.101}\)
\(\Rightarrow\frac{1}{2}A=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{100}-\frac{1}{101}\)
\(\Rightarrow\frac{1}{2}A=1-\frac{1}{101}=\frac{101}{101}-\frac{1}{101}=\frac{100}{101}\)
\(\Rightarrow A=\frac{100}{101}:\frac{1}{2}=\frac{100}{101}.2=\frac{200}{101}=1\frac{99}{101}\)
NH
1
18 tháng 8 2015
\(A=\left(-\frac{2}{3}\right).\left(-\frac{5}{6}\right).\left(-\frac{9}{10}\right)...\left(\frac{-779}{780}\right)=\left(-\frac{4}{6}\right).\left(-\frac{10}{12}\right).\left(-\frac{18}{20}\right)....\left(-\frac{1558}{1560}\right)\)
\(A=\frac{\left(-1.4\right).\left(-2.5\right).\left(-3.6\right)....\left(-38.41\right)}{\left(2.3\right).\left(3.4\right).\left(4.5\right)....\left(39.40\right)}=\frac{\left(1.2.3....38\right).\left(4.5.6...41\right)}{\left(2.3.4...39\right).\left(3.4.5...40\right)}\) ( Vì từ -1 đến -38 có 38 số =>tích của 38 số âm = tích của 38 số dương)
\(A=\frac{\left(1.2.3....38\right).\left(4.5.6...41\right)}{\left(2.3.4...39\right).\left(3.4.5...40\right)}=\frac{1.41}{39.3}=\frac{41}{117}\)
B
0
24 tháng 4 2015
a)\(=\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{5\cdot6}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}\)
\(=\left(1-\frac{1}{6}\right)+\left(\frac{1}{2}-\frac{1}{2}\right)+...+\left(\frac{1}{5}-\frac{1}{5}\right)\)
\(=\left(1-\frac{1}{6}\right)+0+...+0=1-\frac{1}{6}=\frac{6}{6}-\frac{1}{6}=\frac{5}{6}\)
b)\(\frac{3}{2\cdot5}+\frac{3}{5\cdot8}+\frac{3}{8\cdot11}+\frac{3}{11\cdot14}=\frac{1}{2}-\frac{1}{5}+\frac{1}{5}-\frac{1}{8}+\frac{1}{8}-\frac{1}{11}+\frac{1}{11}-\frac{1}{14}\)
\(=\left(\frac{1}{2}-\frac{1}{14}\right)+\left(\frac{1}{5}-\frac{1}{5}\right)+\left(\frac{1}{8}-\frac{1}{8}\right)+\left(\frac{1}{11}-\frac{1}{11}\right)\)
\(=\left(\frac{1}{2}-\frac{1}{14}\right)+0+...+0=\frac{1}{2}-\frac{1}{14}=\frac{7}{14}-\frac{1}{14}=\frac{6}{14}\)
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DP
1 tháng 8 2017
\(C=\frac{3}{1}+\frac{3}{3}+\frac{3}{6}+\frac{3}{10}+....+\frac{3}{5050}\)
\(C=\frac{6}{2}+\frac{6}{6}+\frac{6}{12}+\frac{6}{20}+...+\frac{6}{10100}\)
\(C=\frac{6}{1.2}+\frac{6}{2.3}+\frac{6}{3.4}+\frac{6}{4.5}+....+\frac{6}{100.101}\)
\(C=6.\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{100}-\frac{1}{101}\right)\)
\(C=6.\left(1-\frac{1}{101}\right)=6.\frac{100}{101}=\frac{600}{601}\)
Vậy \(C=\frac{600}{601}\)
\(D=\frac{1}{1.3.5}+\frac{1}{3.5.7}+....+\frac{1}{2009.2011.2013}\)
\( D=\left(\frac{4}{1.3.5}+\frac{4}{3.5.7}+...+\frac{4}{2009.2011.2013}\right)-\left(\frac{1}{1.3.5}+\frac{1}{3.5.7}+...+\frac{1}{2009.2011.2013}\right)\)
\(D=\frac{1}{4}.\left(\frac{1}{1.3}-\frac{1}{3.5}+\frac{1}{3.5}-\frac{1}{5.7}+...+\frac{1}{2009.2011}-\frac{1}{2011.2013}\right)\)
\(D=\frac{1}{4}.\left(\frac{1}{1.3}-\frac{1}{2011.2013}\right)=\frac{1}{4}.\frac{1349380}{4048143}=\frac{1349380}{16192572}\)
Vậy \(D=\frac{1349380}{16192572}\)
10 tháng 8 2016
a) \(\frac{1}{3}.\frac{-6}{13}.\frac{-9}{10}.\frac{-13}{36}\)
\(=\left(\frac{1}{3}.\frac{-9}{10}\right)\left(\frac{-6}{13}.\frac{-13}{36}\right)\)
\(=\frac{-3}{10}.\frac{1}{6}\)
\(=\frac{-1}{20}\)
b) \(\frac{-1}{3}.\frac{-15}{17}.\frac{34}{45}\)
\(=\frac{-1}{3}.\frac{-2}{3}\)
\(=\frac{2}{9}\)
c) \(\left(1-\frac{1}{5}\right)\left(\frac{-3}{10}+\frac{1}{5}\right)\)
\(=\frac{4}{5}.\frac{-1}{10}\)
\(=\frac{-2}{25}\)
d) \(A=\frac{1}{3}.\frac{4}{5}+\frac{1}{3}.\frac{6}{5}+\frac{2}{3}\)
\(=\frac{1}{3}\left(\frac{4}{5}+\frac{6}{5}\right)+\frac{2}{3}\)
\(=\frac{1}{3}.2+\frac{2}{3}\)
\(=\frac{2}{3}+\frac{2}{3}\)
\(=\frac{4}{3}\)
e) \(11\frac{1}{4}-\left(2\frac{5}{7}+5\frac{1}{4}\right)\)
\(=\left(11\frac{1}{4}-5\frac{1}{4}\right)-2\frac{5}{7}\)
\(=6-2\frac{5}{7}\)
\(=5\frac{7}{7}-2\frac{5}{7}\)
\(=3\frac{2}{7}\)