\(A=\frac{1}{1.3.5}+\frac{1}{3.5.7}+...+\frac{1}{25.27.29}\)
\(\Rightarrow4A=\frac{4}{1.3.5}+\frac{4}{3.5.7}+...+\frac{4}{25.27.29}\)
\(=\frac{1}{1.3}-\frac{1}{3.5}+\frac{1}{3.5}-\frac{1}{5.7}+...+\frac{1}{25.27}-\frac{1}{27.29}\)
\(=\frac{1}{3}-\frac{1}{783}=\frac{261}{783}-\frac{1}{783}=\frac{260}{783}\)
A = \(\frac{1}{1.3.5}+\frac{1}{3.5.7}+....+\frac{1}{25.27.29}\)
A = \(\frac{1}{4}.\left(\frac{5-1}{1.3.5}+\frac{7-3}{3.5.7}+...+\frac{29-25}{25.27.29}\right)\)
A = \(\frac{1}{4}.\left(\frac{5}{1.3.5}-\frac{1}{1.3.5}+\frac{7}{3.5.7}-\frac{3}{3.5.7}+...+\frac{29}{25.27.29}-\frac{25}{25.27.29}\right)\)
A = \(\frac{1}{4}.\left(\frac{1}{1.3}-\frac{1}{3.5}+\frac{1}{3.5}-\frac{1}{5.7}+...+\frac{1}{25.27}-\frac{1}{27.29}\right)\)
A = \(\frac{1}{4}.\left(\frac{1}{3}-\frac{1}{27.29}\right)\)
A = \(\frac{1}{4}.\frac{206}{783}\)
A = \(\frac{65}{783}\)
\(4A=\frac{4}{1.3.5}+\frac{4}{3.5.7}+...+\frac{4}{25.27.29}\)
\(4A=\frac{1}{1.3}-\frac{1}{3.5}+\frac{1}{5.7}-\frac{1}{7.9}+...+\frac{1}{25.27}-\frac{1}{27.29}\)
4A = \(\frac{1}{1.3}-\frac{1}{27.29}=\frac{1}{3}-\frac{1}{783}=\frac{260}{783}\)
A = \(\frac{260}{783}:4=\frac{65}{783}\)
