Answer
$a_n=\frac{3}{2}(\frac{2}{3})^{n-1}=(\frac{2}{3})^{n-2}$, $S_{10}=\frac{58025}{13122}$
Work Step by Step
Given the sequence $\frac{3}{2},1,\frac{2}{3},\frac{4}{9},\frac{8}{27},...$, we have $\frac{1}{3/2}-=\frac{2}{3}, \frac{2/3}{1}=\frac{2}{3},...$ and it can be identified as a geometric sequence. We have $a_n=\frac{3}{2}(\frac{2}{3})^{n-1}=(\frac{2}{3})^{n-2}$ and $S_{10}=\frac{3}{2}\times\frac{1-(\frac{2}{3})^{10}}{1-(\frac{2}{3})}=\frac{58025}{13122}$
