Chapter 11 - Sequences; Induction; the Binomial Theorem - Section 11.3 Geometric Sequences; Geometric Series - 11.3 Assess Your Understanding - Page 844: 45

$S_n=1-2^n$

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