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.