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.

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!