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The image contains a question in Arabic. Here is a translated version:

"Which of the following represents the sum of the arithmetic series given by the general formula for the series?"

The options provided are:
1. \( \frac{7(6+1)}{2} \)
2. \( 7(7-1) \)
3. \( \frac{7(7+1)}{2} \)
4. \( 7(7+2) \)

To solve this problem, let's first recall the formula for the sum \( S_n \) of the first \( n \) terms of an arithmetic series:

\[
S_n = \frac{n}{2} \times (a_1 + a_n)
\]

Where:
- \( n \) is the number of terms,
- \( a_1 \) is the first term,
- \( a_n \) is the last term.

Given the formula options, it appears that the formula \( \frac{n(n+1)}{2} \) is the correct general formula for the sum of the first \( n \) positive integers, where \( n \) is the last term.

Thus, the correct answer is:

\[ \boxed{3. \frac{7(7+1)}{2}} \]

This represents the sum of the first 7 natural numbers.

Do you need further details or explanations on this problem?

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Here are 5 related questions that expand on this concept:

1. How can we derive the formula for the sum of the first \( n \) natural numbers?
2. What is the sum of the first 10 terms of an arithmetic sequence where the first term is 2 and the common difference is 3?
3. How would the sum formula change if the series were a geometric series instead of arithmetic?
4. Can the sum formula \( S_n = \frac{n}{2} \times (a_1 + a_n) \) be applied to a series with negative terms?
5. What happens if the arithmetic sequence has a common difference of zero?

**Tip:** When working with sequences, always identify if the sequence is arithmetic, geometric, or another type to use the appropriate formula for the sum or other properties.

Understand how to solve the problem of identifying the sum of an arithmetic series using the general formula. Learn about the formula for the sum of arithmetic series, including detailed steps and examples. Suitable for students in Grades 9-12.

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