Write the sum in sigma notation LaTeX: \displaystyle 12+\frac{180}{29}+\frac{2700}{841}\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{180}{29}}{12}=\frac{15}{29} . Using LaTeX: \displaystyle a_{1} and LaTeX: \displaystyle r to write in sigma notation gives LaTeX: \displaystyle \sum_{n=1}^{\infty}12\left( \frac{15}{29} \right)^{n-1} . Using the formula for the sum of an infinite geometric series gives LaTeX: \displaystyle \frac{12}{1-(\frac{15}{29})}=\frac{174}{7}