《2025年黄冈小状元同步计算天天练六年级数学下册北师大版》

1. 计算：$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac{1}{1024}+\frac{1}{2048}$
想：从两个数开始，逐渐增加加数的个数，寻找规律。
$\frac{1}{2}+\frac{1}{4}=$______ $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}=$______ $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}=$______
后面一个数是前面一个数的$\frac{1}{2}$，而且它们的结果有规律：
$\frac{1}{2}+\frac{1}{4}=1 - \frac{1}{4}=\frac{3}{4}$ $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}=1 - \frac{1}{8}=$______
$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}=$______$-$______$=$______
所以$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac{1}{1024}+\frac{1}{2048}=$______$-$______$=$______

答案： $\frac{3}{4}$ $\frac{7}{8}$ $\frac{15}{16}$ $\frac{7}{8}$
$1 - \frac{1}{16} = \frac{15}{16}$ $1 - \frac{1}{2048} = \frac{2047}{2048}$

2. 计算：$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}$

答案： $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64} + \frac{1}{128} + \frac{1}{256}$
$= 1 - \frac{1}{256}$
$= \frac{255}{256}$

3. 计算：$\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\frac{2}{81}+\frac{2}{243}+\frac{2}{729}$

答案： 想：$\frac{2}{3} + \frac{2}{9} = \frac{8}{9}$
$\frac{2}{3} + \frac{2}{9} + \frac{2}{27} = \frac{26}{27}$
$\frac{2}{3} + \frac{2}{9} = 1 - \frac{1}{9} = \frac{8}{9}$
$\frac{2}{3} + \frac{2}{9} + \frac{2}{27} = 1 - \frac{1}{27} = \frac{26}{27}$
所以 $\frac{2}{3} + \frac{2}{9} + \frac{2}{27} + \frac{2}{81} + \frac{2}{243} + \frac{2}{729}$
$= 1 - \frac{1}{729}$
$= \frac{728}{729}$