given the explicit formula below, write the recursive formula for the sequence g(n)=-12+6(n-1)

To convert the **explicit formula** \( g(n) = -12 + 6(n-1) \) into a **recursive formula**, we follow these steps:

### Explicit Formula:
\[ g(n) = -12 + 6(n-1) \]

Simplify:
\[ g(n) = -12 + 6n - 6 = 6n - 18 \]

### Recursive Formula:
A recursive formula defines each term based on the previous term, \( g(n-1) \).

1. From the explicit formula, observe that the difference between consecutive terms is constant:
   \[
   g(n) - g(n-1) = (6n - 18) - [6(n-1) - 18] = 6
   \]
   This is the common difference.

2. Define the recursive relationship:
   \[
   g(n) = g(n-1) + 6, \quad \text{for } n \geq 2
   \]

3. Include the base case (the first term of the sequence):
   \[
   g(1) = -12
   \]

### Final Recursive Formula:
\[
g(n) = 
\begin{cases} 
-12, & \text{if } n = 1 \\
g(n-1) + 6, & \text{if } n \geq 2
\end{cases}
\]

Would you like me to verify it further or illustrate the sequence? 

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### Related Questions:
1. What is the general difference between explicit and recursive formulas?
2. Can this sequence be represented as a geometric progression?
3. How can you graphically represent this arithmetic sequence?
4. What are the first 10 terms of the sequence, explicitly and recursively?
5. Can you derive the explicit formula from the recursive formula?

### Tip:
For arithmetic sequences, always check the **common difference** when transitioning between recursive and explicit formulas.

Given the explicit formula below, write the recursive formula for the sequence g(n) = -12 + 6(n-1)

Learn how to convert the explicit formula g(n) = -12 + 6(n-1) into a recursive formula. This step-by-step guide explains arithmetic sequences, highlights the common difference, and provides both the recursive formula and base case. Suitable for Grades 9-12.

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