Question

What is the sum of the infinite geometric series?

Answer 1

first of all, let's find the common ratio in this geometric sequence, by simply dividing a any of the terms by the term right behind it, hmmm say -12 and 16

`16~~,~~-12~~,~~9~~,~~-\cfrac{27}{4}~\hfill \stackrel{\textit{\LARGE common ratio}}{\cfrac{-12}{16}\implies -\cfrac{3}{4}} \\\\[-0.35em] ~\dotfill\\\\ \qquad \qquad \textit{sum of an infinite geometric sequence} \\\\ \displaystyle S=\sum\limits_(i=0)^(\infty)\ a_1\cdot r^i\implies S=\cfrac{a_1}{1-r}\quad \begin{cases} a_1=\textit{first term}\\ r=\textit{common ratio}\\ \qquad -1 < r < 1\\[-0.5em] \hrulefill\\ a_1=16\\[1em] r=-(3)/(4) \end{cases}`

`S=\cfrac{16}{ ~~ 1-\left( -(3)/(4) \right) ~~ }\implies S=\cfrac{16}{ ~~ 1 + (3)/(4) ~~ }\implies S=\cfrac{16}{~~ ( 7)/( 4) ~~}\implies S=\cfrac{64}{7}`