a, \(\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{5}{12}+\dfrac{19}{30}\)
\(=\left(\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{5}{12}\right)+\left(\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{19}{30}\right)\)
\(=1+1=2\)
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\(\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+..+\dfrac{1}{1998.1999}+\dfrac{1}{1999.2000}\)
\(=\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{1998}-\dfrac{1}{1999}+\dfrac{1}{1999}-\dfrac{1}{2000}\)
\(=1-\dfrac{1}{2000}=\dfrac{1999}{2000}.\)
\(\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{5}{12}+\dfrac{19}{30}\)
\(=\left(\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{5}{12}\right)+\left(\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{19}{30}\right)\)
\(=\dfrac{12}{12}+\dfrac{30}{30}=1+1=2\)
\(\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+.....+\dfrac{1}{1998.1999}+\dfrac{1}{1999.2000}\)
\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+.....+\dfrac{1}{1998}-\dfrac{1}{1999}+\dfrac{1}{1999}+\dfrac{1}{2000}\)\(=1-\dfrac{1}{2000}=\dfrac{1999}{2000}\)
a) \(\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{5}{12}+\dfrac{19}{30}\)
\(=\dfrac{20}{60}+\dfrac{15}{60}+\dfrac{12}{60}+\dfrac{10}{60}+\dfrac{25}{60}+\dfrac{38}{60}\)
\(=\dfrac{20+15+12+10+25+38}{60}\)
\(=\dfrac{120}{60}\)
\(=2\)
b) \(\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{1998.1999}+\dfrac{1}{1999.2000}\)
\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{1999}-\dfrac{1}{2000}\)
\(=1-\dfrac{1}{2000}\)
\(=\dfrac{1999}{2000}\)
