第29页 - 经纶学典学霸六年级数学上下册【江苏国标】

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$ \frac16×\frac17=\frac16-\frac17$

$\frac{1}{2000×2001}+ \frac{1}{2001×2002} + ...\ $

$+ \frac{1}{2024×2025} + \frac{1}{2025×2026}+\frac{1}{2026}$

$=\frac{1}{2000}-\frac{1}{2001}+\frac{1}{2001}-\frac{1}{2002}+...$

$+\frac{1}{2024}-\frac{1}{2025}+\frac{1}{2025}-\frac{1}{2026}+\frac{1}{2026}$

$=\frac{1}{2000} $

原式

$=1000(1-\frac12+\frac12-\frac13+…+\frac {1}{2025}-\frac {1}{2026})$

$=1000×\frac {2025}{2026}$

$=\frac {1012500}{1013}$

$\ \ \ \ \ \frac{1}{6}+\frac {1}{12}+\frac{1}{20}+\frac{1}{30}+...+\frac {1}{72}+\frac{1}{90}\ $

$=\frac{1}{2×3}+\frac {1}{3×4}+\frac {1}{4×5}+\frac{1}{5×6}+...+\frac{1}{8×9}+\frac{1}{9×10}\ \ $

$=\frac{1}{2}-\frac{1}{3}+\frac13- \frac{1}{4}+\frac14-\frac{1}{5}+\frac15-\frac{1}{6}$

$+...+\frac{1}{8}-\frac{1}{9}+\frac19-\frac{1}{10}$

$=\frac{1}{2}-\frac{1}{10}=\frac{2}{5} $

$\ \ \ \ \frac{1}{3×5}+\frac {1}{5×7}+\frac {1}{7×9}+...+\frac{1}{97×99}\ \ $

$=(\frac{2}{3×5}+\frac{2}{5×7}+\frac{2}{7×9}+...+\frac{2}{97×99})×\frac{1}{2}\ $

$=(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{97}-\frac{1}{99})×\frac{1}{2}\ $

$=(\frac{1}{3}-\frac{1}{99} )×\frac{1}{2}=\frac{32}{99}×\frac{1}{2}$

$=\frac{16}{99} $