Chapter 12 - Sequences; Induction; the Binomial Theorem - 12.3 Geometric Sequences; Geometric Series - 12.3 Assess Your Understanding - Page 824: 45

$1-2^n$

We are given the sum: $-1-2-4-8-....-(2^{n-1})$ Consider the geometric sequence: $a_1=-1$ $r=2$ The given sum represents the sum $S_n$ of the first $n$ terms. Determine this sum: $S_n=a_1\dfrac{1-r^n}{1-r}=(-1)\cdot\dfrac{1-2^n}{1-2}=1-2^n$