a:Khuyến=\(\dfrac{8}{1\cdot5}+\dfrac{8}{5\cdot9}+...+\dfrac{8}{21\cdot25}\)
\(=2\left(\dfrac{4}{1\cdot5}+\dfrac{4}{5\cdot9}+...+\dfrac{4}{21\cdot25}\right)\)
\(=2\left(1-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{9}+...+\dfrac{1}{21}-\dfrac{1}{25}\right)\)
\(=2\left(1-\dfrac{1}{25}\right)=2\cdot\dfrac{24}{25}=\dfrac{48}{25}\)
b:Dũng=\(\dfrac{2}{1\cdot3}+\dfrac{2}{3\cdot5}+...+\dfrac{2}{51\cdot53}\)
\(=1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{51}-\dfrac{1}{53}\)
\(=1-\dfrac{1}{53}=\dfrac{52}{53}\)
c:Bình=\(\dfrac{6}{2\cdot4}+\dfrac{6}{4\cdot6}+...+\dfrac{6}{12\cdot14}\)
\(=3\left(\dfrac{2}{2\cdot4}+\dfrac{2}{4\cdot6}+...+\dfrac{2}{12\cdot14}\right)\)
\(=3\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+...+\dfrac{1}{12}-\dfrac{1}{14}\right)\)
\(=3\left(\dfrac{1}{2}-\dfrac{1}{14}\right)=3\cdot\dfrac{12}{28}=\dfrac{36}{28}=\dfrac{9}{7}\)
d: Nhật=\(\dfrac{1}{3\cdot10}+\dfrac{1}{10\cdot17}+...+\dfrac{1}{31\cdot38}\)
\(=\dfrac{1}{7}\left(\dfrac{7}{3\cdot10}+\dfrac{7}{10\cdot17}+...+\dfrac{7}{31\cdot38}\right)\)
\(=\dfrac{1}{7}\left(\dfrac{1}{3}-\dfrac{1}{10}+\dfrac{1}{10}-\dfrac{1}{17}+...+\dfrac{1}{31}-\dfrac{1}{38}\right)\)
\(=\dfrac{1}{7}\left(\dfrac{1}{3}-\dfrac{1}{38}\right)=\dfrac{1}{7}\cdot\dfrac{35}{114}=\dfrac{5}{114}\)
