a) \(\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{99\cdot100}\)
\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{99}+\dfrac{1}{99}-\dfrac{1}{100}\)
\(=1-\dfrac{1}{100}\)
\(=\dfrac{99}{100}\)
b) \(\dfrac{3}{1\cdot5}+\dfrac{3}{5\cdot9}+\dfrac{3}{9\cdot13}+...+\dfrac{3}{81\cdot85}\)
\(=\dfrac{3}{4}\cdot\left(\dfrac{4}{1\cdot5}+\dfrac{4}{5\cdot9}+\dfrac{4}{9\cdot13}+...+\dfrac{4}{81\cdot85}\right)\)
\(=\dfrac{3}{4}\cdot\left(1-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{13}+\dfrac{1}{13}-\dfrac{1}{81}+\dfrac{1}{81}-\dfrac{1}{85}\right)\)
\(=\dfrac{3}{4}\cdot\left(1-\dfrac{1}{85}\right)\)
\(=\dfrac{3}{4}\cdot\dfrac{84}{85}=\dfrac{63}{85}\)
c) \(\dfrac{2}{3\cdot5}+\dfrac{2}{5\cdot7}+\dfrac{2}{7\cdot9}+...+\dfrac{2}{99\cdot101}\)
\(=1+\left(\dfrac{1}{3\cdot5}+\dfrac{1}{5\cdot7}+\dfrac{1}{7\cdot9}+...+\dfrac{1}{99\cdot101}\right)\)
\(=1\cdot\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{99}+\dfrac{1}{99}-\dfrac{1}{101}\right)\)
\(=1\cdot\left(1-\dfrac{1}{101}\right)\)
\(=1\cdot\dfrac{100}{101}=\dfrac{101}{101}\)
d) \(\dfrac{8}{1\cdot5}+\dfrac{8}{5\cdot9}+\dfrac{8}{9\cdot13}+...+\dfrac{8}{41\cdot45}\)
\(\dfrac{8}{4}\left(\dfrac{4}{1\cdot5}+\dfrac{4}{5\cdot9}+\dfrac{4}{9\cdot13}+...+\dfrac{4}{41\cdot45}\right)\)
\(=\dfrac{8}{4}\cdot\left(1-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{13}+\dfrac{1}{13}-\dfrac{1}{41}+\dfrac{1}{41}-\dfrac{1}{45}\right)\)
\(=\dfrac{8}{4}\cdot\left(1-\dfrac{1}{45}\right)\)
\(=2\cdot\dfrac{44}{45}=\dfrac{88}{45}\)
Sửa lại câu D
d) \(\dfrac{8}{1\cdot5}+\dfrac{8}{5\cdot9}+\dfrac{8}{9\cdot13}+...+\dfrac{8}{41\cdot45}\)
\(=4\cdot\left(\dfrac{4}{1\cdot5}+\dfrac{4}{5\cdot9}+\dfrac{4}{9\cdot13}+...+\dfrac{4}{41\cdot45}\right)\)
\(=4\cdot\left(\dfrac{5}{1\cdot5}-\dfrac{1}{1\cdot5}+\dfrac{9}{5\cdot9}-\dfrac{5}{5\cdot9}+\dfrac{13}{9\cdot13}-\dfrac{9}{9\cdot13}+...+\dfrac{45}{41\cdot45}-\dfrac{41}{41\cdot45}\right)\)
\(=4\cdot\left(1-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{13}+\dfrac{1}{13}-\dfrac{1}{41}+\dfrac{1}{41}-\dfrac{1}{45}\right)\)
\(=4\cdot\left(1-\dfrac{1}{45}\right)\)
\(=4\cdot\dfrac{44}{45}\)
\(=\dfrac{176}{45}\)
