https://file.aiquickdraw.com/mathbot/file/1728648693024paf3q7bt.jpg

In this image, we see a sequence of figures constructed using matchsticks (چوب کبریت), and the problem appears to ask for identifying the number of matchsticks used in each figure and finding a general formula that expresses the relationship between the figure number \( n \) and the number of matchsticks \( a_n \).

Let's break it down:

### Analysis:
- **Figure 1** (شکل 1) uses 5 matchsticks (۵ چوب کبریت).
- **Figure 2** (شکل 2) uses 8 matchsticks (۸ چوب کبریت).
- **Figure 3** (شکل 3) uses 11 matchsticks (۱۱ چوب کبریت).

The number of matchsticks follows an arithmetic sequence. The difference between consecutive terms is constant:
- \( 8 - 5 = 3 \)
- \( 11 - 8 = 3 \)

This suggests that the sequence is linear, meaning the number of matchsticks increases by 3 for each new figure.

### General formula:
The general form of an arithmetic sequence is:
\[
a_n = a_1 + (n - 1) d
\]
Where:
- \( a_1 \) is the first term (number of matchsticks in figure 1),
- \( d \) is the common difference between terms (3 in this case),
- \( n \) is the figure number.

For this sequence:
- \( a_1 = 5 \),
- \( d = 3 \).

Thus, the formula becomes:
\[
a_n = 5 + (n - 1) \times 3
\]
Simplifying:
\[
a_n = 3n + 2
\]

### Filling the table:
- For \( n = 1 \), \( a_1 = 3(1) + 2 = 5 \) (matches with Figure 1).
- For \( n = 2 \), \( a_2 = 3(2) + 2 = 8 \) (matches with Figure 2).
- For \( n = 3 \), \( a_3 = 3(3) + 2 = 11 \) (matches with Figure 3).

This formula can now be used to find the number of matchsticks for any figure \( n \).

Would you like more details or clarifications?

### Related Questions:
1. How can we determine the formula for more complex sequences?
2. What happens if the difference between consecutive terms is not constant?
3. How can we graph this sequence to visualize the relationship?
4. How would the formula change if the first figure used a different number of matchsticks?
5. Can this method be applied to sequences involving shapes other than matchstick figures?

### Tip:
Always check if the difference between terms is constant to identify an arithmetic sequence easily!

Find the relationship between the figure number (n) and the number of matchsticks (a_n) based on the provided sequence of figures constructed using matchsticks.

Explore how to determine the formula for a linear sequence of matchstick figures, where each figure uses an increasing number of matchsticks. The pattern follows an arithmetic sequence, and the general formula for this sequence is a_n = 3n + 2.

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