Chapter 11 - Sequences; Induction; the Binomial Theorem - Section 11.3 Geometric Sequences; Geometric Series - 11.3 Assess Your Understanding - Page 844: 41

$S_n= \dfrac{1}{4}(2^{n}-1)$

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: $a_{1}=\dfrac{1}{4} ; \ r=2$ Now, $S_n= \dfrac{1}{4}\left(\dfrac{1-2^{n}}{1-2}\right) \\=-\dfrac{1}{4}(1-2^{n}) $ Therefore, $S_n= \dfrac{1}{4}(2^{n}-1)$