例 巧算：$\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}$。
▶思路点拨

▶尝试解答

$\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}$
$=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}$
$=1-\frac{1}{6}$
$=\frac{5}{6}$

1. (仿例练习)巧算：$1-\frac{1}{4}+\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+\frac{1}{56}$

$1-\frac{1}{4}+\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+\frac{1}{56}$
$=1-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}$
$=1-\frac{1}{8}$
$=\frac{7}{8}$

2.（变式提升）巧算：$\frac{5}{6} - \frac{7}{12} + \frac{9}{20} - \frac{11}{30} + \frac{13}{42} - \frac{15}{56} + \frac{17}{72}$。

$\frac{5}{6}-\frac{7}{12}+\frac{9}{20}-\frac{11}{30}+\frac{13}{42}-\frac{15}{56}+\frac{17}{72}$
$=(\frac{1}{2}+\frac{1}{3})-(\frac{1}{3}+\frac{1}{4})+(\frac{1}{4}+\frac{1}{5})-(\frac{1}{5}+\frac{1}{6})+(\frac{1}{6}+\frac{1}{7})-(\frac{1}{7}+\frac{1}{8})+(\frac{1}{8}+\frac{1}{9})$
$=\frac{1}{2}+\frac{1}{9}$
$=\frac{11}{18}$

3. (素养培优)巧算：$1+\frac{1}{2}-\frac{5}{6}+\frac{7}{12}-\frac{9}{20}+\frac{11}{30}-\frac{13}{42}$。

$1+\frac{1}{2}-\frac{5}{6}+\frac{7}{12}-\frac{9}{20}+\frac{11}{30}-\frac{13}{42}$
$=1+\frac{1}{2}-(\frac{1}{2}+\frac{1}{3})+(\frac{1}{3}+\frac{1}{4})-(\frac{1}{4}+\frac{1}{5})+(\frac{1}{5}+\frac{1}{6})-(\frac{1}{6}+\frac{1}{7})$
$=1-\frac{1}{7}$
$=\frac{6}{7}$