To find the sum of the terms of a geometric sequence, you can use the formula for the sum of the first \( n \) terms of a geometric sequence:

\[
S_n = a_1 \frac{1 - r^n}{1 - r}
\]

where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( a_1 \) is the first term of the sequence,
- \( r \) is the common ratio,
- \( n \) is the number of terms.

For the given sequence \( 6, 18, 54, 162, \ldots \):
- \( a_1 = 6 \)
- \( r = 3 \)

### Sum of the First \( n \) Terms

Using the formula, the sum of the first \( n \) terms \( S_n \) is:

\[
S_n = 6 \frac{1 - 3^n}{1 - 3}
\]

\[
S_n = 6 \frac{1 - 3^n}{-2}
\]

\[
S_n = -3 (1 - 3^n)
\]

\[
S_n = 3(3^n - 1)
\]

### Example: Sum of the First 4 Terms

Let's calculate the sum of the first 4 terms:

\[
S_4 = 3(3^4 - 1)
\]

\[
S_4 = 3(81 - 1)
\]

\[
S_4 = 3 \times 80
\]

\[
S_4 = 240
\]

So, the sum of the first 4 terms is 240.

### Would you like more details or have any questions?

Here are some related questions you might consider:

1. How can you find the sum of the first 10 terms of this sequence?
2. What is the sum of the first \( n \) terms if the common ratio is a fraction?
3. How does the formula for the sum change if the common ratio \( r \) is negative?
4. What is the sum to infinity for a geometric sequence, and when is it applicable?
5. Can the formula for the sum of a geometric series be derived from first principles?

**Tip:** For geometric sequences where \( |r| < 1 \), the sum to infinity can be calculated using \( S = \frac{a_1}{1 - r} \), which is useful for series that converge.

Learn how to find the sum of terms in the geometric sequence 6, 18, 54, 162, ... using the formula for the sum of the first n terms of a geometric sequence. Detailed calculations and examples provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation