To determine the formula based on the pattern of diamonds in the image, let's go through the analysis step by step.

### Analyzing the Pattern:
1. **Stage 1:** There is 1 diamond.
2. **Stage 2:** There are 3 diamonds.
3. **Stage 3:** There are 5 diamonds.

### Identifying the Sequence:
- The pattern shows that each stage adds 2 more diamonds compared to the previous stage.
- The number of diamonds in each stage forms an arithmetic sequence: \( 1, 3, 5, 7, \dots \)

### Arithmetic Sequence Formula:
For an arithmetic sequence, the general formula for the \(n\)-th term \(a_n\) is:
\[
a_n = a_1 + (n-1) \cdot d
\]
where:
- \(a_1\) is the first term.
- \(d\) is the common difference.

In this case:
- \(a_1 = 1\) (the first term is 1 diamond).
- \(d = 2\) (the difference between each term is 2 diamonds).

Thus, the formula for the number of diamonds at stage \(n\) is:
\[
a_n = 1 + (n-1) \cdot 2 = 2n - 1
\]

### Conclusion:
The formula for the number of diamonds at stage \(n\) is:
\[
\text{Number of diamonds} = 2n - 1
\]

This formula gives you the total number of diamonds in any stage \(n\).

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Would you like more details or have any other questions? Here are some related questions:

1. How would the sequence change if each stage added 3 diamonds instead of 2?
2. Can you derive the formula for the sum of diamonds from stage 1 to \(n\)?
3. What is the 10th term in this sequence?
4. How does this arithmetic sequence differ from a geometric sequence?
5. What are some real-world applications of arithmetic sequences like this one?

**Tip:** Arithmetic sequences often appear in problems involving regular, repeated patterns. Recognizing the common difference is key to solving these types of problems.

Learn how to determine the formula for a diamond pattern sequence using arithmetic sequences. This problem involves identifying the number of diamonds at each stage based on a regular pattern. Suitable for students in Grades 7-9.

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