a)\(\frac{1}{3}+\frac{1}{15}+\frac{1}{35}+...+\frac{1}{195}\)
Đặt \(C=\frac{1}{10}+\frac{1}{15}+\frac{1}{21}+...+\frac{1}{66}\)
\(\Rightarrow\frac{1}{2}C=\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+...+\frac{1}{132}\)
\(\Rightarrow\frac{1}{2}C=\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+...+\frac{1}{11.12}\)
\(\Rightarrow\frac{1}{2}C=\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+...+\frac{1}{11}-\frac{1}{12}\)
\(\Rightarrow\frac{1}{2}C=\frac{1}{4}+\left(-\frac{1}{5}+\frac{1}{5}\right)+\left(-\frac{1}{6}+\frac{1}{6}\right)+...+\left(-\frac{1}{11}+\frac{1}{11}\right)-\frac{1}{12}\)\(\Rightarrow\frac{1}{2}C=\frac{1}{4}+0+0+...+0-\frac{1}{12}\)
\(\Rightarrow\frac{1}{2}C=\frac{1}{4}-\frac{1}{12}\)
\(\Rightarrow\frac{1}{2}C=\frac{3}{12}-\frac{1}{12}\)
\(\Rightarrow\frac{1}{2}C=\frac{2}{12}\)
\(\Rightarrow\frac{1}{2}C=\frac{1}{6}\)
\(\Rightarrow C=\frac{1}{6}:\frac{1}{2}\)
\(\Rightarrow C=\frac{1}{6}\cdot2\)
\(\Rightarrow C=\frac{2}{6}=\frac{1}{3}\)
