Chapter 11 - Section 11.4 - Partial Sums of Arithmetic and Geometric Sequences - Exercise Set - Page 658: 14

$S_{∞}=67.5$

Given geometric sequence $45,15,5,...$ $a_{1}= 45$ Common ratio $r = \frac{a_{n}}{a_{n-1}}$ $r= \frac{a_{2}}{a_{1}} = \frac{15}{45} = \frac{1}{3}$ $|r| \lt 1$, So $S_{∞}$ exists. Sum of the terms of an infinite geometric sequence is $S_{∞}=\frac{a_{1}}{1-r}$ Substituting $a_{1}$ and $r$ $S_{∞}=\frac{45}{1-\frac{1}{3}}$ $S_{∞}=\frac{45}{\frac{2}{3}}$ $S_{∞}=45 \times \frac{3}{2}$ $S_{∞}= \frac{135}{2}$ $S_{∞}=67.5$