Question

What is the sum of the infinite geometric series represented by (picture)

Answer 1

ANSWER

D.
$720$


EXPLANATION

The infinite geometric series given to us is,

$\sum_{n=1}^{\infty}180\times (\frac{3}{4})^{n-1}$


The first term of this series is,



$a_1=180\times (\frac{3}{4})^{1-1}$

This implies that,

$a_1=180\times (\frac{3}{4})^{0}$


$a_1=180$


$a_2=180\times (\frac{3}{4})^{2-1}$


$a_2=180\times (\frac{3}{4})^{1}$



$a_2=135$



The common ratio of this sequence,



$r = \frac{a_2}{a_1}$


$r = \frac{135}{180}$


$r = \frac{3}{4}$


The sum to infinity of this series is given by the formula,

$S_{\infty}=\frac{a_1}{1 - r}$


We substitute the above values to get,

$S_{\infty}=\frac{180}{1 - \frac{3}{4} }$


This simplifies to

$S_{\infty}=\frac{180}{ \frac{1}{4} }$

This implies that,

$S_{\infty}=180 \times 4$


$S_{\infty}=720$

The correct answer is D.