\(\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+.....+\frac{1}{97.100}=\frac{1}{3}\left(\frac{3}{1.4}+\frac{3}{4.7}+\frac{3}{7.10}+....+\frac{3}{97.100}\right)=\frac{1}{3}\left(\frac{1}{1}-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-.......+\frac{1}{97}-\frac{1}{100}\right)=\frac{1}{3}\left(1-\frac{1}{100}\right)=\frac{1}{3}.\frac{99}{100}=\frac{33}{100}\)

Gọi dãy phân số trên là A

A = \(\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+...+\frac{1}{97.100}\)

A = \(1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+...+\frac{1}{97}-\frac{1}{100}\)

A = \(1-\frac{1}{100}\)

A = \(\frac{99}{100}\)

\(\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+...+\frac{1}{97.100}\)

=\(\frac{1}{3}.\left(1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+...+\frac{1}{97}-\frac{1}{100}\right)\)

=\(\frac{1}{3}.\left(1-\frac{1}{100}\right)\)

=\(\frac{1}{3}.\frac{99}{100}\)=\(\frac{33}{100}\)

\(A=\frac{1}{1\cdot4}+\frac{1}{4\cdot7}+...+\frac{1}{91\cdot94}=\frac{1}{3}\left(\frac{3}{1\cdot4}+\frac{3}{4\cdot7}+...+\frac{3}{91\cdot94}\right)\)

\(=\frac{1}{3}\left(1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+...+\frac{1}{91}-\frac{1}{94}\right)\)

\(=\frac{1}{3}\left[\left(1-\frac{1}{94}\right)+\left(\frac{1}{4}-\frac{1}{4}\right)+...+\left(\frac{1}{91}-\frac{1}{91}\right)\right]\)

\(=\frac{1}{3}\left[\left(\frac{94}{94}-\frac{1}{94}\right)+0+...+0\right]=\frac{1}{3}\cdot\frac{93}{94}=\frac{93}{282}\)

Ta thấy: 1/1-1/4 = 3/4 = 3.(1/1.4)

           1/4-1/7 = 3/28 = 3.(1/4.7)

A = 3(1/1-1/4+1/4-1/7+...+1/97-1/100)

A = 3.(1-1/100)

A = 3.(99/100)

A = 297/100

\(A=\frac{1}{3}.\left(1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+.......+\frac{1}{97}-\frac{1}{100}\right)\)

\(A=\frac{1}{3}.\left(1-\frac{1}{100}\right)\)

\(A=\frac{1}{3}.\frac{99}{100}\)

\(A=\frac{33}{100}\)

Võ Thiện Tuấn viết tổng quát kết quả hay phép đề bài hả bạn ?

\(=1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7} +....+\frac{1}{100}-\frac{1}{103}\)

\(=1-\frac{1}{103}\)

\(=\frac{102}{103}\)

Viết tổng quát đề bài ấy \(\frac{1}{n\left(n+3\right)}\)

\(A=\frac{1}{1.4}-\frac{1}{4.7}-\frac{1}{7.10}-...-\frac{1}{2011.2014}\)

\(A=\frac{1}{1.4}-\left(\frac{1}{4.7}+\frac{1}{7.10}+...+\frac{1}{2011.2014}\right)\)

Đặt \(B=\frac{1}{4.7}+\frac{1}{7.10}+...+\frac{1}{2011.2014}\)

\(B=\frac{1}{3}.\left(\frac{3}{4.7}+\frac{3}{7.10}+...+\frac{3}{2011.2014}\right)\)

\(B=\frac{1}{3}.\left(\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+...+\frac{1}{2011}-\frac{1}{2014}\right)\)

\(B=\frac{1}{3}.\left(\frac{1}{4}-\frac{1}{2014}\right)\)

\(B=\frac{1}{3}.\frac{1005}{4028}=\frac{335}{4028}\)

\(A=\frac{1}{4}-\frac{335}{4028}=\frac{168}{1007}\)

A = \(\frac{1}{1.4}-\frac{1}{4.7}-\frac{1}{7.10}-...-\frac{1}{2011.2014}\)

A = 1 + \(\frac{1}{4}\) - \(\frac{1}{4}\) + \(\frac{1}{7}\) - \(\frac{1}{7}\) + \(\frac{1}{10}\) -....- \(\frac{1}{2011}\) + \(\frac{1}{2014}\)

A = 1 + \(\frac{1}{2014}\) = \(\frac{2015}{2014}\)

Sai đề : \(\frac{1}{2011.2014}\)

\(A=\frac{1}{1.4}-\frac{1}{4.7}-\frac{1}{7.10}-...-\frac{1}{2011.2014}\)

\(A=\frac{1}{1.4}-\left(\frac{1}{4.7}+\frac{1}{7.10}+...+\frac{1}{2011.2014}\right)\)

Đặt \(B=\frac{1}{4.7}+\frac{1}{7.10}+...+\frac{1}{2011.2014}\)

\(B=\frac{1}{3}\left(\frac{3}{4.7}+\frac{3}{7.10}+...+\frac{3}{2011.2014}\right)\)

\(B=\frac{1}{3}.\left(\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+...+\frac{1}{2011}-\frac{1}{2014}\right)\)

\(B=\frac{1}{3}.\left(\frac{1}{4}-\frac{1}{2014}\right)\)

\(B=\frac{1}{3}.\frac{1005}{4028}=\frac{335}{4028}\)

\(A=\frac{1}{4}-\frac{335}{4028}=\frac{168}{1007}\)

Chúc bạn học tốt !!!

\(\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+...+\frac{1}{97.100}\)

\(=1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+...+\frac{1}{97}-\frac{1}{100}\)

\(=1-\frac{1}{100}\)

\(=\frac{99}{100}\)

\(\Rightarrow\frac{99}{100}=\frac{0.33.x}{2009}\)

\(\Rightarrow100.0.33.x=99.2009\)

\(\Rightarrow0x=198891\Rightarrow\)không có GT x thỏa mãn

Ta có : 1/ 1.4 + 1/ 4.7 + .... + 1/ 2016.2019 .

      = 1 - 1/4 + 1/4 - 1/7 + ... + 1/2016 - 1/2019 .

      = 1 - 1/2019 .

      = 2018/2019 .

\(\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+...+\frac{1}{2016.2019}\)

\(=\frac{1}{3}.\left(\frac{3}{1.4}+\frac{3}{4.7}+\frac{3}{7.10}+...+\frac{3}{2016.2019}\right)\)

\(=\frac{1}{3}.\left(1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+...+\frac{1}{2016}-\frac{1}{2019}\right)\)

\(=\frac{1}{3}.\left(1-\frac{1}{2019}\right)\)

\(=\frac{1}{3}.\frac{2018}{2019}\)

\(=\frac{2018}{6057}\)

_Chúc bạn học tốt_