# Chapter 11 - Section 11.4 - Partial Sums of Arithmetic and Geometric Sequences - Exercise Set - Page 658: 16

$S_{∞}=\frac{4}{5}$

#### Work Step by Step

Given geometric sequence $\frac{3}{5},\frac{3}{20},\frac{3}{80},...$ $a_{1}= \frac{3}{5}$ Common ratio $r = \frac{a_{n}}{a_{n-1}}$ $r= \frac{a_{2}}{a_{1}} =\frac{\frac{3}{20}}{\frac{3}{5}} = \frac{3}{20} \times \frac{5}{3} = \frac{1}{4}$ $|r| \lt 1$, So $S_{∞}$ exists. Sum of the terms of an infinite geometric sequence is $S_{∞}=\frac{a_{1}}{1-r}$ Substituting $a_{1}$ and $r$ $S_{∞}=\frac{\frac{3}{5}}{1-\frac{1}{4}}$ $S_{∞}=\frac{\frac{3}{5}}{\frac{3}{4}}$ $S_{∞}=\frac{3}{5} \times \frac{4}{3}$ $S_{∞}= \frac{4}{5}$

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