The problem involves finding an explicit formula for a recursively defined sequence. Here are the given details:

• $a_1 = -3$
• $a_n = 6 \cdot a_{n-1}$ for $n \geq 2$

Step-by-Step Solution:

1. Analyze the Recursive Formula: The recursive formula indicates that each term is $6$ times the previous term. This suggests the sequence grows (or shrinks) geometrically.

2. Find the Explicit Formula: In a geometric sequence, the general term is given by: $a_n = a_1 \cdot r^{n-1}$ where $a_1$ is the first term and $r$ is the common ratio.

   Here:

   • $a_1 = -3$
   • $r = 6$

   Substituting these values: $a_n = -3 \cdot 6^{n-1}$

Final Explicit Formula:

$a_n = -3 \cdot 6^{n-1}$

Would you like further clarification or additional examples?

Here are 5 questions related to this problem for deeper understanding:

1. How does the recursive formula determine the pattern in the sequence?
2. What would the explicit formula be if the first term were different, say $a_1 = 2$?
3. How do geometric sequences differ from arithmetic sequences?
4. Can you calculate $a_5$ using both the recursive and explicit formulas?
5. How would the formula change if the common ratio were a fraction?

Tip: To verify your explicit formula, calculate the first few terms using both the recursive and explicit formulas to ensure they match.