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.