母题1 计算下面各题。
(1) $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64}$
(2) $\frac{1}{100} + \frac{2}{100} + \frac{3}{100} + \dots + \frac{99}{100}$
图解思路:第(1)题可以利用数形结合的思想解答,如。第(2)题中各个分数的分母相同,只需依次把分子相加,分母不变。
规范解答:
(1) $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64}$
$= \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64} + \frac{1}{64} - \frac{1}{64}$
$= 1 - \frac{1}{64}$
$= \frac{63}{64}$
(2) $\frac{1}{100} + \frac{2}{100} + \frac{3}{100} + \dots + \frac{99}{100}$
$= \frac{1 + 2 + 3 + \dots + 99}{100}$
$= \frac{(1 + 99) × 99 ÷ 2}{100}$
$= 49.5$

(1) $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64}$
$= \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64} + \frac{1}{64} - \frac{1}{64}$
$= 1 - \frac{1}{64}$
$= \frac{63}{64}$
(2) $\frac{1}{100} + \frac{2}{100} + \frac{3}{100} + \dots + \frac{99}{100}$
$= \frac{1 + 2 + 3 + \dots + 99}{100}$
$= \frac{(1 + 99) × 99 ÷ 2}{100}$
$= 49.5$

1 分母是20的所有真分数的和是()，分母是20的所有最简真分数的和是()。

1. $\frac{19}{2}$，4

2 计算下面各题。
(1) $\frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{15}{16} + \frac{31}{32} + \frac{63}{64} + \frac{127}{128}$
(2) $\frac{1}{1000} + \frac{3}{1000} + \frac{5}{1000} + \dots + \frac{999}{1000}$

(1) $\frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{15}{16} + \frac{31}{32} + \frac{63}{64} + \frac{127}{128}$
$=(1 - \frac{1}{2}) + (1 - \frac{1}{4}) + (1 - \frac{1}{8}) + (1 - \frac{1}{16}) + (1 - \frac{1}{32}) + (1 - \frac{1}{64}) + (1 - \frac{1}{128})$
$=7 - (\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64} + \frac{1}{128})$
$=7 - (1 - \frac{1}{128})$
$=7 - 1 + \frac{1}{128}$
$=6 \frac{1}{128}$
(2) $\frac{1}{1000} + \frac{3}{1000} + \frac{5}{1000} + \dots + \frac{999}{1000}$
$= \frac{1 + 3 + 5 + \dots + 999}{1000}$
$= \frac{(1 + 999) × 500 ÷ 2}{1000}$
$=250$