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2022-05-01 21:42:05
\(\dfrac{1}{2}+\dfrac{5}{6}+\dfrac{11}{12}+\dfrac{19}{20}+\dfrac{29}{30}+\dfrac{41}{42}+\dfrac{55}{56}+\dfrac{71}{72}+\dfrac{89}{90}=1-\dfrac{1}{2}+1-\dfrac{1}{6}+1-\dfrac{1}{12}+....+1-\dfrac{1}{90}=1+1+...+1-\left(\dfrac{1}{2}+\dfrac{1}{6}+...+\dfrac{1}{90}\right)=9-\left(\dfrac{1}{1x2}+\dfrac{1}{2x3}+...+\dfrac{1}{9x10}\right)=9-\left(\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+....+\dfrac{1}{9}-\dfrac{1}{10}\right)=9-\left(1-\dfrac{1}{10}\right)=9-\dfrac{9}{10}=\dfrac{81}{10}\)