To find a recursive formula for the sequence $\{12, 36, 108, 324, 972, \dots\}$, we first need to identify the pattern in the sequence.

1. First, let's look at the ratio between consecutive terms:
   • $\frac{36}{12} = 3$
   • $\frac{108}{36} = 3$
   • $\frac{324}{108} = 3$
   • $\frac{972}{324} = 3$

So, each term is multiplied by 3 to get the next term. This suggests that the sequence follows a geometric progression.

Recursive Formula:

Given that the sequence follows a geometric progression with a common ratio of 3, the recursive formula for the sequence is:

$a_1 = 12$ $a_n = 3 \cdot a_{n-1}, \quad \text{for} \quad n \geq 2$

Thus, the sequence starts with $a_1 = 12$, and each subsequent term is obtained by multiplying the previous term by 3.

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Would you like a detailed explanation of how to derive the recursive formula, or do you have any other questions? Here are some related questions:

1. What is the explicit (closed-form) formula for this sequence?
2. How would you find the nth term of a geometric sequence?
3. How can you solve for the nth term if you're given the recursive formula?
4. How does the common ratio in a geometric sequence affect the terms?
5. How do you derive a recursive formula for an arithmetic sequence?

Tip: When working with geometric sequences, always look for the common ratio between consecutive terms to identify the pattern!