A = \(\frac{1}{2.4}+\frac{1}{4.6}+....+\frac{1}{2012.2014}\)
A = \(\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+....+\frac{1}{2012}-\frac{1}{2014}\right)\)
A = \(\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{2014}\right)\)
A = \(\frac{1}{2}.\frac{503}{1007}\)
A = \(\frac{503}{2014}\)
2A=\(\frac{4-2}{2.4}+\frac{6-4}{4.6}+...+\frac{2014-2012}{2012.2014}\)
\(2A=\frac{4}{2.4}-\frac{2}{2.4}+\frac{6}{4.6}-\frac{4}{4.6}+...+\frac{2014}{2012.2014}-\frac{2012}{2012.2014}\)
\(2A=\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+...+\frac{1}{2012}-\frac{1}{2014}\)
\(2A=\frac{1}{2}-\frac{1}{2014}=\frac{503}{1007}\Rightarrow A=\frac{503}{2014}\)
\(A=\frac{1}{2.4}+\frac{1}{4.6}+...+\frac{1}{2012.2014}\)
\(\Rightarrow2A=2\left(\frac{1}{2.4}+\frac{1}{4.6}+...+\frac{1}{2012.2014}\right)\)
\(\Rightarrow2A=\frac{2}{2.4}+\frac{2}{4.6}+...+\frac{2}{2012.2014}\)
\(\Rightarrow2A=\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+...+\frac{1}{2012}-\frac{1}{2014}\)
\(\Rightarrow2A=\frac{1}{2}-\frac{1}{2014}\)
\(\Rightarrow2A=\frac{503}{1007}\)
\(\Rightarrow A=\frac{503}{2014}\)
\(A=\frac{1}{2.4}+\frac{1}{4.6}+..........+\frac{1}{2012.2014}\)
\(2A=\frac{2}{2.4}+\frac{2}{4.6}+.....+\frac{2}{2012.2014}\)
\(2A=\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+......+\frac{1}{2012}-\frac{1}{2014}\)
\(2A=\frac{1}{2}-\frac{1}{2014}\)
\(2A=\frac{503}{1007}\)=> \(A=\frac{503}{2014}\)
