What is the sum of the first seven terms of the geometric series?

Ask by Harmon Vega. in the United States

Jan 21,2025

Question

What is the sum of the first seven terms of the geometric series? $$ \begin{array}{l}3+12+48+192+\ldots \ 4,372 \ 12,288 \ 16,383 \ 65,535\end{array} $$

UpStudy AI · Accepted Answer

To find the sum of the first seven terms of the geometric series $$ 3 + 12 + 48 + 192 + \ldots $$, follow these steps:

1. **Identify the First Term and Common Ratio:**
   - **First term ($$ a $$):** 3
   - **Common ratio ($$ r $$):** $$ \frac{12}{3} = 4 $$

2. **Use the Formula for the Sum of the First $$ n $$ Terms of a Geometric Series:**
   $$
   S_n = a \frac{r^n - 1}{r - 1}
   $$
   Where:
   - $$ S_n $$ is the sum of the first $$ n $$ terms.
   - $$ a $$ is the first term.
   - $$ r $$ is the common ratio.
   - $$ n $$ is the number of terms.

3. **Plug in the Values:**
   $$
   S_7 = 3 \times \frac{4^7 - 1}{4 - 1}
   $$
   $$
   S_7 = 3 \times \frac{16,384 - 1}{3}
   $$
   $$
   S_7 = 3 \times \frac{16,383}{3} = 16,383
   $$

**Answer:** 16,383
The sum of the first seven terms is 16,383.

What is the sum of the first seven terms of the geometric series?

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