Find the sum of the first four terms of the geometric sequence, 6, 18, 54, 162...

The given sequence is geometric, with the first term \( a = 6 \) and the common ratio \( r = \frac{18}{6} = 3 \).

The sum of the first \( n \) terms of a geometric sequence is given by the formula:

\[
S_n = \frac{a(1 - r^n)}{1 - r}
\]

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.

For the first four terms (\( n = 4 \)):

\[
S_4 = \frac{6(1 - 3^4)}{1 - 3}
\]

Now let's calculate this:

\[
S_4 = \frac{6(1 - 81)}{1 - 3} = \frac{6(-80)}{-2} = \frac{-480}{-2} = 240
\]

Thus, the sum of the first four terms of the sequence is \( S_4 = 240 \).

Would you like more details on geometric sequences or have any questions?

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Here are 5 related questions to explore:
1. How do you derive the formula for the sum of a geometric sequence?
2. What happens to the sum if the common ratio is less than 1?
3. Can you find the sum of the first 5 terms of this sequence?
4. What is the sum of an infinite geometric series if \( |r| < 1 \)?
5. How does the common ratio affect the rate of growth of the sequence?

**Tip**: In a geometric sequence, if \( |r| > 1 \), the terms grow rapidly; if \( |r| < 1 \), the terms decrease toward zero.

Discover how to find the sum of the first four terms of a geometric sequence starting with 6, 18, 54, 162. This step-by-step guide covers key concepts like geometric sequences and the sum formula, making it perfect for students in Grades 8-10.

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