Chapter 11 - Sequences; Induction; the Binomial Theorem - Section 11.3 Geometric Sequences; Geometric Series - 11.3 Assess Your Understanding - Page 844: 42

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

The sum of the First $n$ Terms of a Geometric Sequence is given by: $S_{n}=a_{1} (\dfrac{1-r^{n}}{1-r}) ; \ r\neq 0,1$ We are given: $a_{1}=\dfrac{3}{9}=\dfrac{1}{3} ; \ r=3$ Now, $S_n= \dfrac{1}{3} \ (\dfrac{1-3^{n}}{1- 3} \ ) \\=-\dfrac{1}{2} \times \dfrac{1}{3} \times (3^{n}-1) $ Therefore, $S_n= \dfrac{1}{6}(3^{n}-1)$