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A = \(\dfrac{1}{3\times6}\) + \(\dfrac{1}{6\times9}\) + \(\dfrac{1}{9\times12}\)+...+\(\dfrac{1}{144\times147}\)
A = \(\dfrac{1}{3}\) \(\times\)( \(\dfrac{3}{3\times6}\) + \(\dfrac{3}{6\times9}\)+\(\dfrac{1}{9\times12}\)+...+\(\dfrac{3}{144\times147}\))
A = \(\dfrac{1}{3}\) \(\times\)(\(\dfrac{1}{3}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{12}+...+\dfrac{1}{144}-\dfrac{1}{147}\))
A = \(\dfrac{1}{3}\)\(\times\)(\(\dfrac{1}{3}\) - \(\dfrac{1}{147}\))
A = \(\dfrac{1}{3}\) \(\times\)\(\dfrac{16}{49}\)
A = \(\dfrac{16}{147}\)
\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{8.9.10}=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}\right)+\frac{1}{2}.\left(\frac{1}{2.3}-\frac{1}{3.4}\right)+...+\frac{1}{2}.\left(\frac{1}{8.9}-\frac{1}{9.10}\right)\)
\(=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{8.9}-\frac{1}{9.10}\right)\)
\(\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{9.10}\right)=\frac{1}{2}.\frac{22}{45}=\frac{11}{45}\)
\(\frac{4}{3.6}+\frac{4}{6.9}+\frac{4}{9.12}+\frac{4}{12.15}\)
\(=\frac{4}{3}.\left(\frac{3}{3.6}+\frac{3}{6.9}+\frac{3}{9.12}+\frac{3}{12.15}\right)\)
\(=\frac{4}{3}.\left(\frac{1}{3}-\frac{1}{6}+\frac{1}{6}-\frac{1}{9}+\frac{1}{9}-\frac{1}{12}+\frac{1}{12}-\frac{1}{15}\right)\)
\(=\frac{4}{3}.\left(\frac{1}{3}-\frac{1}{15}\right)\)
\(=\frac{4}{3}.\left(\frac{5}{15}-\frac{1}{15}\right)\)
\(=\frac{4}{3}.\frac{4}{15}=\frac{16}{45}\)
Dấu . là nhân nha
\(\frac{4}{3.6}+\frac{4}{6.9}+\frac{4}{9.12}+\frac{4}{12.15}\)
\(=\frac{4}{3}.\left(\frac{3}{3.6}+\frac{3}{6.9}+\frac{3}{9.12}+\frac{3}{12.15}\right)\)
\(=\frac{4}{3}.\left(\frac{1}{3}-\frac{1}{6}+\frac{1}{6}-\frac{1}{9}+\frac{1}{9}-\frac{1}{12}+\frac{1}{12}-\frac{1}{15}\right)\)
\(=\frac{4}{3}.\left(\frac{1}{3}-\frac{1}{15}\right)\)
\(=\frac{4}{3}.\frac{4}{15}=\frac{16}{45}\)
\(A=\frac{1}{1\times6\times6}+\frac{1}{2\times9\times8}+\frac{1}{3\times12\times10}+...+\frac{1}{98\times297\times200}\)
\(A=\frac{1}{1\times\left(2\times3\right)\times\left(2\times3\right)}+\frac{1}{2\times\left(3\times3\right)\times\left(2\times4\right)}+...+\frac{1}{98\times\left(99\times3\right)\times\left(100\times2\right)}\)
\(A=\frac{1}{6}\times\left(\frac{1}{1\times2\times3}+\frac{1}{2\times3\times4}+...+\frac{1}{98\times99\times100}\right)\)
\(12\times A=\frac{2}{1\times2\times3}+\frac{2}{2\times3\times4}+...+\frac{2}{98\times99\times100}\)
\(12\times A=\left(\frac{1}{1\times2}-\frac{1}{2\times3}\right)+\left(\frac{1}{2\times3}-\frac{1}{3\times4}\right)+...+\left(\frac{1}{98\times99}-\frac{1}{99\times100}\right)\)
\(12\times A=\frac{1}{1\times2}-\frac{1}{99\times100}=\frac{4949}{9900}\)
\(A=\frac{4949}{118800}\)
\(x\times\left(\frac{2015}{8\times9}+\frac{1925}{9\times10}+\frac{1795}{10\times11}+\frac{1629}{11\times12}+6\right)=\frac{1}{24}\)
=> \(x\times\left(\frac{2015}{72}+\frac{1925}{90}+\frac{1795}{110}+\frac{1629}{132}+6\right)=\frac{1}{24}\)
=> \(x\times84\frac{3}{88}=\frac{1}{24}\Rightarrow x=\frac{1}{24}:\frac{7395}{88}=\frac{1}{24}\times\frac{88}{7395}=\frac{88}{177480}=\frac{11}{22185}\)
\(\frac{1.2.6.4.6.4.5.2}{2.3.6.8.6.2.2.2.8.10}=\frac{1}{96}\)
\(=\frac{1}{2}\times\left(\frac{2}{2\times4}+\frac{2}{4\times6}+...+\frac{2}{12\times14}\right)\)
\(=\frac{1}{2}\times\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+...+\frac{1}{12}-\frac{1}{14}\right)\)
\(=\frac{1}{2}\times\left(\frac{1}{2}-\frac{1}{14}\right)=\frac{1}{2}\times\frac{3}{7}=\frac{3}{14}\)
\(\frac{1}{2.6}+\frac{1}{4.9}+\frac{1}{6.12}+...+\frac{1}{36.57}+\frac{1}{38.60}\)
\(=\frac{1}{2.3}.\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{18.19}+\frac{1}{19.20}\right)\)
\(=\frac{1}{6}.\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{18}-\frac{1}{19}+\frac{1}{19}-\frac{1}{20}\right)\)
\(=\frac{1}{6}.\left(1-\frac{1}{20}\right)\)
\(=\frac{1}{6}.\frac{19}{20}=\frac{19}{120}\)
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