Question

find the sum of the first 8 terms of the following geometric sequence:

2, 6, 18, 54, 162, ...

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, \(a = 2\), to find \(r\), we divide the second term by the first term: \(r=\frac{6}{2}=3\). The number of terms \(n = 8\).

Step2: Substitute into the formula

The formula for the sum of the first \(n\) terms of a geometric sequence is \(S=\frac{a(1 - r^{n})}{1 - r}\). Substituting \(a = 2\), \(r = 3\), 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:

The sum of the first 8 terms is \(6560\).