Chapter 11 - 11.3 - Geometric Sequences and Series - 11.3 Exercises - Page 796: 80

$\frac{7}{22}$

$0.3181818...=0.3+0.018+0.00018+0.0000018+...=0.3+S=\frac{3}{10}+S$ $S=0.018+0.00018+0.0000018+...=0.018(0.01)^0+0.018(0.01)^1+0.018(0.01)^2+...$ $a_1=0.018$ $r=\frac{a_2}{a_1}=\frac{0.018(0.01)^1}{0.018(0.01)^0}=0.01$ $S=\frac{a_1}{1-r}=\frac{0.018}{1-0.01}=\frac{0.018}{0.99}=\frac{18}{990}=\frac{1}{55}$ $0.3181818...=\frac{3}{10}+\frac{1}{55}=\frac{33+2}{110}=\frac{35}{110}=\frac{7}{22}$