solve for s10=3-3(-2)raise to 10 use geometric series

To solve for \( s_{10} \) using a geometric series, we need to clarify the components. The general formula for the sum of the first \( n \) terms of a geometric series is:

\[
S_n = a \cdot \frac{1 - r^n}{1 - r} \quad \text{(if \( r \neq 1 \))}
\]

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.

Given the expression:

\[
s_{10} = 3 - 3(-2)^{10}
\]

We interpret this as potentially involving a geometric series where the common ratio \( r = -2 \). Let's break down the steps:

1. **Find the first term ( \( a \) )**: The first term, \( a \), is 3 (based on the equation provided).
2. **Find the common ratio ( \( r \) )**: The common ratio \( r = -2 \), based on the powers of \( -2 \).
3. **Number of terms ( \( n = 10 \) )**: The problem asks for \( s_{10} \), so we are summing the first 10 terms.

Now, using the formula for the sum of a geometric series, substitute the values:

\[
S_{10} = 3 \cdot \frac{1 - (-2)^{10}}{1 - (-2)}
\]

Let's calculate this.

\[
S_{10} = 3 \cdot \frac{1 - 1024}{1 + 2} = 3 \cdot \frac{1 - 1024}{3} = 3 \cdot \frac{-1023}{3} = 3 \cdot (-341) = -1023
\]

Thus, the sum of the first 10 terms is:

\[
s_{10} = -1023
\]

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Do you want any further details or have any questions?

### 5 Related Questions:
1. How does the common ratio affect the behavior of a geometric series?
2. What happens to the sum of a geometric series when the common ratio \( |r| > 1 \)?
3. Can the sum formula for an infinite geometric series be used here? Why or why not?
4. How would the sum change if \( r \) were positive?
5. What is the difference between an arithmetic and a geometric series?

### Tip:
Always verify the common ratio carefully when working with geometric series, as it strongly influences whether the terms grow, shrink, or alternate.

Explore how to solve the problem s10 = 3 - 3(-2)^10 using the geometric series formula. This detailed solution includes concepts like common ratio, exponents, and the sum of a geometric series. Suitable for students in Grades 9-12.

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