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 the geometric sequence 6, 18, 54, 162 using the geometric series sum formula. This step-by-step solution is ideal for students in Grades 9-11, covering key concepts like geometric sequences and series summation.

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