To determine the recursive formula for the given sequence $2, 6, 18, 54, 162, 486$, let's observe the pattern:

Step 1: Analyze the pattern

The ratio between consecutive terms is constant: $\frac{6}{2} = 3, \quad \frac{18}{6} = 3, \quad \frac{54}{18} = 3, \quad \frac{162}{54} = 3, \quad \frac{486}{162} = 3$

This indicates that the sequence is geometric with a common ratio of 3.

Step 2: Define the recursive formula

The first term of the sequence is: $a_1 = 2$ For $n > 1$, each term is obtained by multiplying the previous term by 3: $a_n = 3 \cdot a_{n-1}, \quad \text{for } n > 1$

Final Recursive Formula:

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

---

Would you like further details on geometric sequences or examples? Here are some related questions to deepen understanding:

1. How do you derive the general (explicit) formula for a geometric sequence?
2. What is the sum of the first $n$ terms of a geometric sequence?
3. How can you identify a recursive formula for a sequence with varying patterns?
4. Can this recursive formula be modified for sequences starting with different values?
5. What are practical applications of geometric sequences?

Tip: When working with sequences, always check for consistency in the differences or ratios between terms to identify patterns!