https://file.aiquickdraw.com/mathbot/file/1727371984512oc95374z.jpg

The image shows a summation (Σ) notation:

\[
\sum_{n=2}^{47} 40(0.96)^{n-2}
\]

This is a geometric series where the terms are given by \( 40(0.96)^{n-2} \), and the summation runs from \( n = 2 \) to \( n = 47 \).

### Step-by-step breakdown:
1. **General term**: The general term of the series is \( 40(0.96)^{n-2} \).
2. **Number of terms**: The sum goes from \( n = 2 \) to \( n = 47 \), so there are \( 47 - 2 + 1 = 46 \) terms in the series.
3. **First term**: The first term when \( n = 2 \) is \( 40(0.96)^{2-2} = 40(0.96)^0 = 40 \).
4. **Common ratio**: The common ratio \( r \) is \( 0.96 \), since each term is multiplied by \( 0.96 \) as \( n \) increases.

This series is geometric, and the sum of a geometric series can be computed using the formula:

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

Where:
- \( a \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the number of terms.

Let's compute the sum using the provided values:

- \( a = 40 \),
- \( r = 0.96 \),
- \( n = 46 \).

I'll calculate the sum.The sum of the geometric series is approximately \( 847.08 \).

Would you like further details or have any questions about the calculation?

Here are five related questions that might interest you:
1. How does the formula for the sum of a geometric series change if the common ratio \( r \) is greater than 1?
2. What happens if the common ratio \( r \) is negative?
3. Can you compute the sum of an infinite geometric series?
4. How does changing the number of terms \( n \) affect the total sum?
5. Could this series be used to model real-world problems, such as depreciation or discounting?

**Tip:** In a geometric series, if \( |r| < 1 \), the series converges as \( n \) approaches infinity, and the sum can be calculated using a simpler formula for infinite series.

Evaluate the summation \( \sum_{n=2}^{47} 40(0.96)^{n-2} \)

Solve the geometric series summation \( \sum_{n=2}^{47} 40(0.96)^{n-2} \) with step-by-step instructions. Learn how to apply the geometric series formula and explore its applications. Suitable for high school students in Grades 10-12.

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