College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 8, Sequences and Series - Section 8.3 - Geometric Sequences - 8.3 Exercises: 50

Answer

$S_4 = \dfrac{80}{81}$

Work Step by Step

RECALL: The partial sum $S_n$ (sum of the first $n$ terms) of a geometric sequence is given by the formula: $S_n=a\left(\dfrac{1-r^n}{1-r}\right), r\ne 1$ where $a$ = first term $r$ = common ratio The given geometric sequence has: $a=\dfrac{2}{3} \\r=\dfrac{1}{3}$ Thus, the sum of the first 4 terms is equal to: $\require{cancel} S_4 =\dfrac{2}{3}\left(\dfrac{1-(\frac{1}{3})^4}{1-\frac{1}{3}}\right) \\S_4=\dfrac{2}{3}\left(\dfrac{1-\frac{1}{81}}{\frac{2}{3}}\right) \\S_4=\dfrac{2}{3}\left(\dfrac{\frac{80}{81}}{\frac{2}{3}}\right) \\S_4=\dfrac{2}{3} \cdot \dfrac{80}{81} \cdot \dfrac{3}{2} \\S_4=\dfrac{\cancel{2}}{\cancel{3}} \cdot \dfrac{80}{81} \cdot \dfrac{\cancel{3}}{\cancel{2}} \\S_4 = \dfrac{80}{81}$
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