a. \(\dfrac{5}{4.6}+\dfrac{5}{6.8}+\dfrac{5}{8.10}+...+\dfrac{5}{298.300}\)
\(=\dfrac{5.2}{4.6.2}+\dfrac{5.2}{6.8.2}+\dfrac{5.2}{8.10.2}+...+\dfrac{5.2}{298.300.2}\)
\(=\dfrac{5}{2}\cdot\left(\dfrac{2}{4.6}+\dfrac{2}{6.8}+\dfrac{2}{8.10}+...+\dfrac{2}{298.300}\right)\)
\(=\dfrac{5}{2}\cdot\left(\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{8}+\dfrac{1}{8}-\dfrac{1}{10}+...+\dfrac{1}{298}-\dfrac{1}{300}\right)\)
\(=\dfrac{5}{2}\cdot\left(\dfrac{1}{4}-\dfrac{1}{300}\right)\)
\(=\dfrac{5}{2}\cdot\dfrac{37}{150}=\dfrac{37}{60}\)
\(=\dfrac{5}{2}\left(\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{8}+...+\dfrac{1}{298}-\dfrac{1}{300}\right)\)
\(=\dfrac{5}{2}\cdot\dfrac{150-2}{600}=\dfrac{1}{120}\cdot74=\dfrac{74}{120}=\dfrac{37}{60}\)
\(\dfrac{5}{4.6}+\dfrac{5}{6.8}+\dfrac{5}{8.10}+...+\dfrac{5}{298.300}\)
\(=5\left(\dfrac{2}{4.6}+\dfrac{2}{6.8}+\dfrac{2}{8.10}+...+\dfrac{2}{298.300}\right)\)
\(=5\left(\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{8}+\dfrac{1}{8}-\dfrac{1}{10}+...+\dfrac{1}{298}-\dfrac{1}{300}\right)\)
\(=5\left(\dfrac{1}{4}-\dfrac{1}{300}\right)\)
\(=5.\dfrac{75-1}{300}=\dfrac{74}{60}=\dfrac{37}{30}\)
