https://file.aiquickdraw.com/mathbot/file/172533615119002bzog8n.jpg

Let's analyze and solve the first problem from the image.

The first problem is asking for the sum \( S_6 \), where the first term \( a = 4 \), and the common difference \( d = -2 \).

This appears to be an arithmetic series, where the sum of the first \( n \) terms \( S_n \) is given by the formula:

\[
S_n = \frac{n}{2} \times [2a + (n-1) \times d]
\]

For \( S_6 \):
- \( n = 6 \)
- \( a = 4 \)
- \( d = -2 \)

Substitute these values into the formula:

\[
S_6 = \frac{6}{2} \times [2(4) + (6-1) \times (-2)]
\]

Simplifying:

\[
S_6 = 3 \times [8 + 5 \times (-2)]
\]
\[
S_6 = 3 \times [8 - 10]
\]
\[
S_6 = 3 \times (-2)
\]
\[
S_6 = -6
\]

So, the sum \( S_6 = -6 \).

Would you like more details on this solution or help with the second problem? Here are some related questions:

1. How do you determine the common difference in an arithmetic series?
2. What happens to the sum if the common difference is positive?
3. Can you derive the formula for the sum of an arithmetic series?
4. What would be the sum of a series where \( a = 4 \) and \( d = 2 \) for \( n = 6 \)?
5. How does the sign of the common difference affect the terms of the series?

**Tip:** Understanding how to manipulate the formula for the sum of an arithmetic series will help you solve these problems more quickly.

Learn how to find the sum of an arithmetic series using the formula. Solve the problem where the first term a = 4 and the common difference d = -2, for n = 6. Detailed step-by-step solution included.

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