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第67页

信息发布者：

$ \begin {aligned}=&(1+3+5+\dots +15+17)+(\frac {1}{6}+\frac {1}{12}+\frac {1}{20}+\dots +\frac {1}{72}+\frac {1}{90})\\=&81+(\frac {1}{2}-\frac {1}{3}+\frac {1}{3}-\frac {1}{4}+\frac {1}{4}-\frac {1}{5}+\dots +\frac {1}{8}-\frac {1}{9}+\frac {1}{9}-\frac {1}{10})\\=&81+(\frac {1}{2}-\frac {1}{10})\\=&81+\frac {2}{5}\\=&81\frac {2}{5}\end {aligned}$

$ \begin {aligned}=&1-\frac {1}{2}+1-\frac {1}{6}+1-\frac {1}{12}+\dots +1-\frac {1}{90}\\=&1×9-(\frac {1}{2}+\frac {1}{6}+\frac {1}{12}+\dots +\frac {1}{90})\\=&9-(1-\frac {1}{2}+\frac {1}{2}-\frac {1}{3}+\frac {1}{3}-\frac {1}{4}+\dots +\frac {1}{9}-\frac {1}{10})\\=&9-(1-\frac {1}{10})\\=&9-1+\frac {1}{10}\\=&8\frac {1}{10}\end {aligned}$

$ \begin {aligned}=&1-\frac {1}{2}+1-\frac {1}{4}+1-\frac {1}{8}+1-\frac {1}{16}+1-\frac {1}{32}+1-\frac {1}{64}+1-\frac {1}{128}\\=&(1+1+1+1+1+1+1)-(\frac {1}{2}+\frac {1}{4}+\frac {1}{8}+\frac {1}{16}+\frac {1}{32}+\frac {1}{64}+\frac {1}{128})\\=&7-(1-\frac {1}{2}+\frac {1}{2}-\frac {1}{4}+\frac {1}{4}-\frac {1}{8}+\frac {1}{8}-\frac {1}{16}+\frac {1}{16}-\frac {1}{32}+\frac {1}{32}-\frac {1}{64}+\frac {1}{64}-\frac {1}{128})\\=&7-(1-\frac {1}{128})\\=&7-1+\frac {1}{128}\\=&6\frac {1}{128}\end {aligned}$

 $\begin{aligned}\frac{1}{24}&=\frac{1×(1+2+3)}{24×(1+2+3)}\\&=\frac{1+2+3}{24×6}\\&=\frac{1+2+3}{144}\\&=\frac{1}{144}+\frac{2}{144}+\frac{3}{144}\\&=\frac{1}{144}+\frac{1}{72}+\frac{1}{48}\end{aligned}$

 答：$M$、$N$、$Q$分别是144、72、48（答案不唯一）。

 $\begin{aligned}&\frac{2}{3}+\frac{1}{6}+\frac{1}{9}+\frac{1}{12}+\frac{1}{15}+\frac{1}{18}-1\\=&\frac{2}{3}+\frac{1}{6}+\frac{2}{18}+\frac{1}{18}+\frac{1}{12}+\frac{1}{15}-1\\=&\frac{2}{3}+\frac{1}{6}+\frac{1}{6}+\frac{1}{12}+\frac{1}{15}-1\\=&\frac{2}{3}+\frac{1}{3}+\frac{1}{12}+\frac{1}{15}-1\\=&1+\frac{1}{12}+\frac{1}{15}-1\\=&\frac{1}{12}+\frac{1}{15}\end{aligned}$

 答：去掉$\frac{1}{12}$和$\frac{1}{15}$这两个分数，才能使余下的分数之和为1。

解：

$\begin {aligned}\frac {△}{11}+\frac {◯}{3}&=\frac {17}{33}\\\frac {3△}{33}+\frac {11◯}{33}&=\frac {17}{33}\\3△+11◯&=17\end {aligned}$

因为△和◯均表示不同的自然数，

所以$△=2，$$◯=1。$

$ △+◯=2+1=3$

答：$△$与$◯$的和是$3。$