Question

представить в виде геометрической дроби 0,(6)         1,2(1)         0,1(7)       0,(35)

Answer 1

$0,(6)= \frac{6}{10} + \frac{6}{100} + \frac{6}{1000} +\cdots, \\ b_1=\frac{6}{10}, q=\frac{1}{10}, \\ S=\frac{b_1}{1-q}=\frac{\frac{6}{10}}{1-\frac{1}{10}}=\frac{6\cdot10}{10\cdot9}=\frac{6}{9}=\frac{2}{3}, \\ 0,(6)=\frac{2}{3};$

$0,1(7)= \frac{1}{10} + \frac{7}{100} + \frac{7}{1000} +\cdots, \\ b_1=\frac{7}{100}, q=\frac{1}{10}, \\ S=\frac{b_1}{1-q}=\frac{\frac{7}{100}}{1-\frac{1}{10}}=\frac{7\cdot10}{100\cdot9}=\frac{7}{90}, \\ 0,1(7)=\frac{1}{10}+\frac{7}{90}= \frac{16}{90};$

$0,(35)= \frac{35}{100} + \frac{35}{10000} + \cdots, \\ b_1=\frac{35}{100}, q=\frac{1}{100}, \\ S=\frac{b_1}{1-q}=\frac{\frac{35}{100}}{1-\frac{1}{100}}=\frac{35\cdot100}{100\cdot99}=\frac{35}{99}, \\ 0,(35)=\frac{35}{99}.$