Chapter 12 - Sequences; Induction; the Binomial Theorem - 12.3 Geometric Sequences; Geometric Series - 12.3 Assess Your Understanding - Page 824: 44

$2(3^n-1)$

We are given the sum: $\sum_{k=1}^n 4\cdot 3^{k-1}$ Consider the geometric sequence: $a_1=4\cdot 3^{1-1}=4$ $r=3$ The given sum represents the sum $S_n$ of the first $n$ terms. Determine this sum: $S_n=a_1\dfrac{1-r^n}{1-r}=4\cdot\dfrac{1-3^n}{1-3}$ $=2(3^n-1)$