Chapter 11 - Sequences; Induction; the Binomial Theorem - Section 11.3 Geometric Sequences; Geometric Series - 11.3 Assess Your Understanding - Page 844: 44

$S_n=2 (3^n-1)$

The sum of the first $n$ terms of a Geometric Sequence is given by: $S_{n}=\displaystyle\sum_{k=1}^n a_1r^{k-1}=a_{1} (\dfrac{1-r^{n}}{1-r}) ; \ r\neq 0,1$ We are given: $a_{1}= 4 ; \ r= 3$ Now, $S_n= 4 \ [\dfrac{1-(3)^{n}}{1- 3} \ ] \\= 4 \ [\dfrac{1-(3)^{n}}{-2} \ ] $ Therefore, $S_n=2 (3^n-1)$