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第13页

信息发布者：

 $1-\dfrac{1}{20}-\dfrac{1}{30}-\dfrac{1}{42}-\dfrac{1}{56}-\dfrac{1}{72}-\dfrac{1}{90}$

 $=1-(\dfrac{1}{4}-\dfrac{1}{5})-(\dfrac{1}{5}-\dfrac{1}{6})-(\dfrac{1}{6}-\dfrac{1}{7})-(\dfrac{1}{7}-\dfrac{1}{8})-(\dfrac{1}{8}-\dfrac{1}{9})-(\dfrac{1}{9}-\dfrac{1}{10})$

 $=1-\dfrac{1}{4}+\dfrac{1}{5}-\dfrac{1}{5}+\dfrac{1}{6}-\dfrac{1}{6}+\dfrac{1}{7}-\dfrac{1}{7}+\dfrac{1}{8}-\dfrac{1}{8}+\dfrac{1}{9}-\dfrac{1}{9}+\dfrac{1}{10}$

 $=1-\dfrac{1}{4}+\dfrac{1}{10}$

 $=\dfrac{3}{4}+\dfrac{1}{10}$

 $=\dfrac{17}{20}$

 $\dfrac{1}{2}+\dfrac{5}{6}+\dfrac{11}{12}+\dots+\dfrac{379}{380}$

 $=(1-\dfrac{1}{2})+(1-\dfrac{1}{6})+(1-\dfrac{1}{12})+\dots+(1-\dfrac{1}{380})$

 $=19-(\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+\dots+\dfrac{1}{380})$

 $=19-(\dfrac{1}{1×2}+\dfrac{1}{2×3}+\dfrac{1}{3×4}+\dots+\dfrac{1}{19×20})$

 $=19-(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dots+\dfrac{1}{19}-\dfrac{1}{20})$

 $=19-(1-\dfrac{1}{20})$

 $=19-1+\dfrac{1}{20}$

 $=18\dfrac{1}{20}$