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We are given a geometric sequence where \( a_1 = 2 \) and the common ratio \( r = -6 \). We need to find the sum of the first five terms.

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

\[
S_n = \frac{a_1(1 - r^n)}{1 - r} \quad \text{(for } r \neq 1 \text{)}
\]

For this problem:
- \( a_1 = 2 \)
- \( r = -6 \)
- \( n = 5 \)

Substituting these values into the formula:

\[
S_5 = \frac{2(1 - (-6)^5)}{1 - (-6)}
\]

Now, I'll calculate the exact sum.The sum of the first five terms of the geometric sequence is \( 2,222 \).

Would you like further details on this solution, or do you have any other questions?

Here are 5 related questions to consider:
1. How would the sum change if the common ratio \( r \) were positive?
2. What is the formula for the sum if \( n \) is not given and we have to compute indefinitely many terms?
3. How does changing \( a_1 \) affect the sum of the sequence?
4. Can you explain the difference between arithmetic and geometric sequences?
5. What would happen if the common ratio \( r = 1 \)? 

**Tip**: When working with geometric sequences, if the absolute value of the ratio \( |r| \) is greater than 1, the terms grow (or shrink) rapidly.

Find the sum of the first five terms of the geometric sequence with a₁ = 2 and r = -6.

Find the sum of the first five terms of a geometric sequence where the first term is 2 and the common ratio is -6. Learn how to apply the sum formula for geometric sequences, including step-by-step calculations. Suitable for high school students in Grades 9-12.

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