Answer
$S_{9}=11,999.999988$
Work Step by Step
We are asked to find the sum of:
$10800+1080+108+...+0.000108$
We see that this is a geometric sequence with $a_1=10800$. We find $r$:
$r=1080/10800=0.1$
We know that a geometric sequence has the form:
$a_{n}=ar^{n-1}$
We use this to find $n$:
$a_n=(10800)(0.1)^{n-1}$
$0.000108=(10800)(0.1)^{n-1}$
$10^{-8}=(0.1)^{n-1}$
$10^{-8}=(10)^{-(n-1)}$
$10^{-8}=(10)^{1-n}$
$-8=1-n$
$n=1+8=9$
We know the partial sum of a geometric sequence is:
$S_n=a_1\frac{1-r^n}{1-r}$
$S_{9}=10800 \frac{1-(\frac{1}{10})^{9}}{1-\frac{1}{10}}=11,999.999988$