\(A=\dfrac{4}{2.4}+\dfrac{4}{4.6}+\dfrac{4}{6.8}+...+\dfrac{4}{2012.2014}=2\left(\dfrac{2}{2.4}+\dfrac{2}{4.6}+\dfrac{2}{6.8}+...+\dfrac{2}{2012.2014}\right)=2\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+...+\dfrac{1}{2012}-\dfrac{1}{2014}\right)=2\left(\dfrac{1}{2}-\dfrac{1}{2014}\right)=2.\dfrac{503}{1007}=\dfrac{1006}{1007}\)
\(A=2\left(\dfrac{2}{2.4}+\dfrac{2}{4.6}+\dfrac{2}{6.8}+...+\dfrac{2}{2012.2012}\right)\)
\(=2\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+...+\dfrac{1}{2012}-\dfrac{1}{2014}\right)\)
\(=2\left(\dfrac{1}{2}+\dfrac{1}{2014}\right)\)
\(=2.\dfrac{504}{1007}\)
\(=\dfrac{1008}{1007}\)
