The given sequence is: $2, 6, 18, 54, 162, 486, \dots$

To find a recursive formula, observe the pattern:

• $\frac{6}{2} = 3$
• $\frac{18}{6} = 3$
• $\frac{54}{18} = 3$
• $\frac{162}{54} = 3$
• $\frac{486}{162} = 3$

This shows that each term is obtained by multiplying the previous term by 3. Therefore, the recursive formula can be expressed as:

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

with the initial condition:

$a_1 = 2$

Thus, the recursive formula for the sequence is:

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

Would you like a more detailed explanation or have any questions?

Here are 5 related questions to expand on this concept:

1. How can we generalize a recursive formula for any geometric sequence?
2. What is the explicit (non-recursive) formula for this sequence?
3. How do you determine whether a sequence is arithmetic or geometric?
4. Can you provide an example of a recursive sequence that decreases over time?
5. What is the sum of the first $n$ terms in a geometric sequence?

Tip: In geometric sequences, the ratio between consecutive terms (called the common ratio) is constant, which makes finding recursive formulas straightforward.