Question

find the sum of the first 7 terms of the following geometric sequence: 3, 15, 75, 375, 1875, ... 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, \(a = 3\), \(n = 7\). To find \(r\), divide the second term by the first term: \(r=\frac{15}{3}=5\).

Step2: Substitute into the formula

Use the sum formula for a geometric sequence \(S=\frac{a(1 - r^{n})}{1 - r}\). Substitute \(a = 3\), \(r = 5\), and \(n = 7\) into the formula:
\[

$$\begin{align*} S&=\frac{3(1 - 5^{7})}{1 - 5}\\ &=\frac{3(1 - 78125)}{-4}\\ &=\frac{3(-78124)}{-4}\\ &=\frac{-234372}{-4}\\ &= 58593 \end{align*}$$

\]

Answer:

\(58593\)