Question

find the sum of the first 8 terms of the following geometric sequence: 2, 6, 18, 54, ... hint: $s = \frac{a(1 - r^n)}{1 - r}$

Explanation:

Step1: Identify \(a\), \(r\), and \(n\)

In a geometric sequence, \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. Here, \(a = 2\), \(r=\frac{6}{2}=3\), and \(n = 8\).

Step2: Substitute into the formula

Use the sum formula for a geometric sequence \(S=\frac{a(1 - r^{n})}{1 - r}\). Substitute \(a = 2\), \(r = 3\), and \(n = 8\) into the formula:
\[

$$\begin{align*} S&=\frac{2(1 - 3^{8})}{1 - 3}\\ &=\frac{2(1 - 6561)}{-2}\\ &=\frac{2(-6560)}{-2}\\ &= 6560 \end{align*}$$

\]

Answer:

\(6560\)