1/1x2 + 1/2x3+ 1/3x4+....+1/2009x2010
= 1/1-1/2 + 1/2-1/3+ 1/3-1/4+...+1/2009-1/2010
= 1/1-1/2010
= 2009/2010
\(\frac{1}{1\cdot2}+\cdot\cdot\cdot+\frac{1}{2009\cdot2010}\)
\(=1-\frac{1}{2}+\cdot\cdot\cdot+\frac{1}{2009}-\frac{1}{2010}\)
\(=1-\frac{1}{2010}\)
\(=\frac{2009}{2010}\)
2/ 3x5+ 2/5x7+ 2/7x9+...+2/2007x2011+ 2/2011x2013
= 1/3-1/5+ 1/5-1/7+ 1/7-1/9+...+1/2007-1/2011+ 1/2011- 1/2013
= 1/3-1/2013
= 670/2013
Ta có:
\(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{2009.2010}\)
= \(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2009}-\frac{1}{2010}\)
= \(1-\frac{1}{2010}=\frac{2009}{2010}\)
\(\frac{2}{3.5}+\frac{2}{5.7}+\frac{2}{7.9}+....+\frac{2}{2009.2011}+\frac{2}{2011.2013}\)(xem lại đề)
= \(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{2009}-\frac{1}{2011}+\frac{1}{2011}-\frac{1}{2013}\)
= \(\frac{1}{3}-\frac{1}{2013}=\frac{670}{2013}\)
\(\frac{1}{1.2}+....+\frac{1}{2009.2010}\)
\(=1-\frac{1}{2}+...+\frac{1}{2009}-\frac{1}{1010}\)
\(=1-\frac{1}{1010}\)
\(=\frac{2009}{2010}\)
mấy câu kia tương tự
\(\frac{2}{3\times5}+\cdot\cdot\cdot+\frac{2}{2011\times2013}\)
\(=\frac{1}{3}-\frac{1}{5}+\cdot\cdot\cdot+\frac{1}{2011}-\frac{1}{2013}\)
\(=\frac{1}{3}-\frac{1}{2013}\)
\(=\frac{670}{2013}\)
\(\frac{1}{1\times4}+\cdot\cdot\cdot+\frac{1}{316\times319}\)
\(=\left(\frac{3}{1\times4}+\cdot\cdot\cdot+\frac{3}{316\times319}\right):3\)
\(=\left(1-\frac{1}{4}+\cdot\cdot\cdot+\frac{1}{316}-\frac{1}{319}\right):3\)
\(=\left(1-\frac{1}{319}\right):3\)
\(=\frac{318}{319}:3\)
\(=\frac{106}{319}\)
\(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2009.2010}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2009}-\frac{1}{2010}\)
\(=1-\frac{1}{2010}\)
\(=\frac{2010-1}{2010}=\frac{2019}{2010}\)
\(\frac{2}{3.5}+\frac{2}{5.7}+\frac{2}{7.9}+...+\frac{2}{2009.2011}+\frac{2}{2011.2013}\)
\(=\frac{2}{3}-\frac{2}{5}+\frac{2}{5}-\frac{2}{7}+\frac{2}{7}-\frac{2}{9}+...+\frac{2}{2009}-\frac{2}{2011}+\frac{2}{2011}-\frac{2}{2013}\)
\(=\frac{2}{3}-\frac{2}{2013}\)
\(=\frac{1342-2}{2013}=\frac{1340}{2013}\)
\(\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+...+\frac{1}{316.319}\)
\(=\left(\frac{3}{1.4}+\frac{3}{4.7}+...+\frac{3}{316.319}\right):3\)
\(=\left(3-\frac{3}{4}+\frac{3}{4}-\frac{3}{7}+\frac{3}{7}-\frac{3}{10}+...+\frac{3}{316}-\frac{3}{319}\right):3\)
\(=\left(3-\frac{3}{319}\right):3\)
\(=\frac{954}{319}:3\)
\(=\frac{318}{319}\)
