Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 12 - Sequences; Induction; the Binomial Theorem - 12.3 Geometric Sequences; Geometric Series - 12.3 Assess Your Understanding - Page 824: 42

Answer

$\dfrac{1}{6}(3^n-1)$

Work Step by Step

We are given the sum: $\dfrac{3}{9}+\dfrac{3^2}{9}+\dfrac{3^3}{9}+....+\dfrac{3^n}{9}$ Consider the geometric sequence: $a_1=\dfrac{3}{9}$ $r=3$ The given sum represents the sum $S_n$ of the first $n$ terms. Determine this sum: $S_n=a_1\dfrac{1-r^n}{1-r}=\dfrac{3}{9}\cdot\dfrac{1-3^n}{1-3}$ $=\dfrac{1}{6}(3^n-1)$
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