《2025年重点中学招生分班五年真题分类卷数学》

16. (2024陕西TYZ)$\frac {1}{1.5^{2}-0.5^{2}}+\frac {1}{2.5^{2}-0.5^{2}}+\frac {1}{3.5^{2}-0.5^{2}}+... +\frac {1}{49.5^{2}-0.5^{2}}+\frac {1}{50.5^{2}-0.5^{2}}$

答案： 解:原式=$\frac{1}{(1.5+0.5)(1.5-0.5)}+\frac{1}{(2.5+0.5)(2.5-0.5)}+\frac{1}{(3.5+0.5)(3.5-0.5)}+…+\frac{1}{(49.5+0.5)(49.5-0.5)}+\frac{1}{(50.5+0.5)(50.5-0.5)}$
=$\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}+…+\frac{1}{49×50}+\frac{1}{50×51}$
=1-$\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+…+\frac{1}{49}-\frac{1}{50}+\frac{1}{50}-\frac{1}{51}$
=1-$\frac{1}{51}$
=$\frac{50}{51}$

17. (2024陕西TYZBH学校)$(\frac {4}{3}-\frac {7}{12}+\frac {11}{28}-\frac {1}{56}-\frac {19}{88}+\frac {30}{209})×19$

答案： 解:原式=[(1+$\frac{1}{3})-(\frac{1}{3}+\frac{1}{4})+(\frac{1}{4}+\frac{1}{7})-(\frac{1}{7}-\frac{1}{8})-(\frac{1}{8}+\frac{1}{11})+(\frac{1}{11}+\frac{1}{19})$]×19
=(1+$\frac{1}{3}-\frac{1}{3}-\frac{1}{4}+\frac{1}{4}+\frac{1}{7}-\frac{1}{7}+\frac{1}{8}-\frac{1}{8}-\frac{1}{11}+\frac{1}{11}+\frac{1}{19}$)×19
=$\frac{20}{19}$×19
=20

18. (2022陕西GDFZ)$[7\frac {1}{3}-(\frac {49}{12}-\frac {63}{20}+\frac {77}{30}-\frac {91}{42}+\frac {105}{56})]÷32$

答案： 解:原式=[7$\frac{1}{3}$-7×($\frac{7}{12}-\frac{9}{20}+\frac{11}{30}-\frac{13}{42}+\frac{15}{56}$)]×$\frac{1}{32}$
=[7$\frac{1}{3}$-7×($\frac{1}{3}+\frac{1}{4}-\frac{1}{4}-\frac{1}{5}+\frac{1}{5}+\frac{1}{6}-\frac{1}{6}-\frac{1}{7}+\frac{1}{7}+\frac{1}{8}$)]×$\frac{1}{32}$
=(7$\frac{1}{3}$-7×$\frac{11}{24}$)×$\frac{1}{32}$
=$\frac{33}{8}$×$\frac{1}{32}$
=$\frac{33}{256}$

19. (2024陕西GDFZ)$\frac {1}{2}+\frac {5}{6}+\frac {11}{12}+\frac {19}{20}+\frac {29}{30}+\frac {41}{42}+\frac {55}{56}+\frac {71}{72}+\frac {89}{90}+\frac {109}{110}$

答案： 解:原式=1-$\frac{1}{2}+1-\frac{1}{6}+1-\frac{1}{12}+…+1-\frac{1}{110}$
=1+1+1+…+1-($\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+…+\frac{1}{110}$)
=1×10-($\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}+…+\frac{1}{10×11}$)
=10-(1-$\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+…+\frac{1}{10}-\frac{1}{11}$)
=10-(1-$\frac{1}{11}$)
=9$\frac{1}{11}$