Chapter 11 - Infinite Sequences and Series - 11.2 Exercises - Page 736: 54

$10.135135135= \frac{5017}{495}$

$10.135135135$ $=10.1 + \frac{35}{10^{3}} + \frac{35}{10^{5}}+\frac{35}{10^{7}}+...$ $a=\frac{35}{10^{3}}$ and $r= \frac{1}{10^{2}}$ $10.135135135 =10.1+ \frac{\frac{35}{10^{3}}}{1-\frac{1}{10^{2}}}$ $=10.1 + \frac{\frac{35}{1000}}{\frac{99}{100}}$ $=\frac{101}{10} + \frac{35}{990}$ $= \frac{10034}{990}=\frac{5017}{495}$