Write the sum in sigma notation $LaTeX: \displaystyle 10+\frac{20}{3}+\frac{40}{9}\ldots$ and then find the sum.

Taking the ratio of the first two terms gives $LaTeX: \displaystyle r = \frac{a_2}{a_1}=\frac{\frac{20}{3}}{10}=\frac{2}{3}$ . Using $LaTeX: \displaystyle a_{1}$ and $LaTeX: \displaystyle r$ to write in sigma notation gives $LaTeX: \displaystyle \sum_{n=1}^{\infty}10\left( \frac{2}{3} \right)^{n-1}$ . Using the formula for the sum of an infinite geometric series gives $LaTeX: \displaystyle \frac{10}{1-(\frac{2}{3})}=30$