Ta có : \(A=\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+...+\frac{1}{2014.2017}=\frac{1}{3}.\left(\frac{3}{1.4}+\frac{3}{4.7}+\frac{3}{7.10}+...+\frac{3}{2014.2017}\right)\)
\(=\frac{1}{3}.\left(1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+...+\frac{1}{2014}-\frac{1}{2017}\right)=\frac{1}{3}.\left(1-\frac{1}{2017}\right)=\frac{1}{3}.\frac{2016}{2017}\)
\(=\frac{672}{2017}\)
\(A=\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+...+\frac{1}{2014.2017}\)
\(A=\frac{1}{3}.\left(1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+...+\frac{1}{2014}-\frac{1}{2017}\right)\)
\(A=\frac{1}{3}.\left(1-\frac{1}{2017}\right)\)
\(A=\frac{1}{3}.\frac{2016}{2017}\)
\(A=\frac{672}{2017}\)