Answer
$7.1234512345... = \frac{237446}{33333}$
Work Step by Step
$7.1234512345...$
$= 7 + \frac{12345}{10^{5}}+ \frac{12345}{10^{10}}+ \frac{12345}{10^{15}}+...$
$= 7+ \Sigma^{\infty}_{n=0} + \frac{12345}{10^{5}} (\frac{1}{10^{5}})^{n}$
$a=\frac{12345}{105}$ and $r=\frac{1}{105}$
$7.1234512345... = 7 + \frac{\frac{12345}{10^{5}}}{1-\frac{1}{10^{5}}}$
$= 7 + \frac{12345}{99999}$
$= 7 + \frac{4115}{33333}$
$= \frac{237446}{33333}$
