\(A=\frac{1}{2.4}+\frac{1}{4.6}+\frac{1}{6.8}+...+\frac{1}{2012.2014}\)
\(\Leftrightarrow A=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+\frac{1}{6}-\frac{1}{8}+...+\frac{1}{2012}-\frac{1}{2014}\right)\)
\(\Leftrightarrow A=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{2014}\right)\)
\(\Leftrightarrow A=\frac{1}{2}\cdot\frac{503}{1007}\)
\(\Leftrightarrow A=\frac{503}{2014}\)
= 1/2[1/2 - 1/4+1/4-1/6 + 1/6-1/8+...+ 1/2012-1/2014]
= 1/2[1/2-1/2014]
= 1/2 * 503/1007
= 503/2014
\(A=\frac{1}{2\cdot4}+\frac{1}{4\cdot6}+\frac{1}{6\cdot8}+...+\frac{1}{2012\cdot2014}\)
\(\Rightarrow A=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+...+\frac{1}{2012}-\frac{1}{2014}\right)\)
\(\Rightarrow A=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{2014}\right)\)
\(\Rightarrow A=\frac{1}{2}\cdot\frac{503}{1007}\)
\(\Rightarrow A=\frac{503}{2014}\)
\(A=\frac{1}{2.4}+\frac{1}{4.6}+\frac{1}{6.8}+....+\frac{1}{2012.2014}\)
\(A=\frac{4-2}{2.4}+\frac{6-4}{4.6}+\frac{8-6}{6.8}+...+\frac{2014-2012}{2012.2014}\)
\(A=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+\frac{1}{6}-\frac{1}{8}+...+\frac{1}{2012}-\frac{1}{2014}\right)\)
\(A=\frac{1}{2}\cdot\left(\frac{1}{2}-\frac{1}{2014}\right)\)
\(A=\frac{1}{2}\cdot\frac{503}{1007}=\frac{503}{2014}\)
Đặt A = \(\frac{1}{2.4}+\frac{1}{4.6}+\frac{1}{6.8}+...+\frac{1}{2012.2014}\)
\(=1-\left(\frac{1}{2}-\frac{1}{4}\right)+\left(\frac{1}{4}-\frac{1}{6}\right)+\left(\frac{1}{6}-\frac{1}{8}\right)+...+\left(\frac{1}{2012}-\frac{1}{2014}\right)\)
\(=1-\frac{1}{2014}\)
\(=\frac{2013}{2014}\)