Đặt \(A=\dfrac{1}{2.4}+\dfrac{1}{4.6}+...+\dfrac{1}{2016.2018}\)
\(\Rightarrow2.A=\dfrac{2}{2.4}+\dfrac{2}{4.6}+...+\dfrac{2}{2016.2018}\)
\(2A=\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+...+\dfrac{1}{2016}-\dfrac{1}{2018}\)
\(2A=\dfrac{1}{2}-\left(\dfrac{1}{4}-\dfrac{1}{4}\right)-\left(\dfrac{1}{6}-\dfrac{1}{6}\right)-...-\left(\dfrac{1}{2016}-\dfrac{1}{2016}\right)-\dfrac{1}{2018}\)
\(2A=\dfrac{1}{2}-0-0-...-0-\dfrac{1}{2018}=\dfrac{1}{2}-\dfrac{1}{2018}\)
\(\Leftrightarrow2A=\dfrac{504}{1009}\Leftrightarrow A=\dfrac{252}{1009}\)
Vậy \(A=\dfrac{252}{1009}\)
1/2.4 +1/4.6+ 1/6.8+........+1/2016.2018
=1/2-1/4+1/4-1/6+1/6-1/8+...+1/2016-1/2018
=1/2+(-1/4+1/4)+(-1/6+1/6)+(-1/8+1/8)+...+(-1/2016+1/2016)+(-1/2018)
=1/2+0+0+0+....+0+(-1/2018)
=1/2+(-1/2018)
=1009/2018+(-1/2018)
=1008/2018
