https://file.aiquickdraw.com/mathbot/file/1729622157423fyhdxr1d.jpg

Let's analyze the given problem step-by-step.

### **Explicit formula:**  
\[ a_n = -2 + (n - 1) \cdot 3 \]  
This formula describes an **arithmetic sequence** where:
- The first term \(a_1\) is obtained by substituting \(n = 1\):
  \[ a_1 = -2 + (1 - 1) \cdot 3 = -2 \]
- The common difference \(d\) is 3 (since each successive term increases by 3).

### **Identifying the Equivalent Recursive Formula:**

A recursive formula for an arithmetic sequence typically has the following form:
\[
a_1 = \text{initial term}, \quad a_n = a_{n-1} + d
\]
Where:
- \(a_1\) is the first term of the sequence, and  
- \(d\) is the common difference between terms.

From our explicit formula:
- \(a_1 = -2\)
- \(d = 3\)

Now let’s compare this with the provided options.

### **Option Analysis:**
- **A**:  
  \[ a_1 = 3, \quad a_n = (a_{n-1} + 3) \cdot 2 \]  
  This is incorrect because the first term \(a_1 = 3\) doesn’t match \(a_1 = -2\), and the common difference formula is also incorrect (it's multiplied by 2).
  
- **B**:  
  \[ a_1 = -2, \quad a_n = a_{n-1} - 3 \]  
  This is incorrect. Although \(a_1 = -2\) is correct, the common difference here is \(-3\) instead of \(3\).

- **C**:  
  \[ a_1 = 3, \quad a_n = a_{n-1} - 2 \]  
  This is incorrect because the first term \(a_1\) is not \(-2\), and the common difference is \(-2\).

- **D**:  
  \[ a_1 = -2, \quad a_n = a_{n-1} + 3 \]  
  This is **correct**! The first term matches \(a_1 = -2\), and the common difference \(d = 3\) is correct.

### **Correct Answer:**
The correct option is **D**. 

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Do you have any further questions or need more details?  
Here are some related questions to extend your understanding:

1. What is the general form of an arithmetic sequence?
2. How do you convert between explicit and recursive formulas?
3. How do we determine the common difference from a sequence?
4. Can an arithmetic sequence have negative common differences?
5. How would the sequence change if the initial term were different?

**Tip:** Always check the first term and the common difference when matching recursive and explicit formulas!

Which of the following recursive formulas represents the same arithmetic sequence as the explicit formula a_n = -2 + (n - 1)3?

Learn how to match recursive formulas with explicit formulas in arithmetic sequences. This question explores the relationship between recursive and explicit expressions, focusing on the sequence a_n = -2 + (n - 1) * 3. Suitable for high school students in Grades 9-12.

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