Certainly! Let's solve the problem using the formula for the nth term of a geometric sequence:

### Given:

- First term (\( a_1 \)) = 324
- We need to find the second and third terms (\( a_2 \) and \( a_3 \)).
- Fourth term (\( a_4 \)) = -12
- The common ratio is \( r \).

### Formula:

\[
a_n = a_1 \cdot r^{n-1}
\]

### Step-by-Step Solution:

1. **Express \( a_4 \) using the formula:**

   Since \( a_4 = -12 \), we have:
   \[
   a_4 = a_1 \cdot r^{4-1} = 324 \cdot r^3 = -12
   \]

2. **Solve for \( r \):**
   \[
   r^3 = \frac{-12}{324} = -\frac{1}{27}
   \]
   Taking the cube root of both sides:
   \[
   r = -\frac{1}{3}
   \]

3. **Find \( a_2 \) and \( a_3 \):**

   - For \( a_2 \):
     \[
     a_2 = a_1 \cdot r^{2-1} = 324 \cdot r = 324 \cdot -\frac{1}{3} = -108
     \]
   - For \( a_3 \):
     \[
     a_3 = a_1 \cdot r^{3-1} = 324 \cdot r^2 = 324 \cdot \left(-\frac{1}{3}\right)^2 = 324 \cdot \frac{1}{9} = 36
     \]

### Conclusion:
Using the formula \( a_n = a_1 \cdot r^{n-1} \), the two geometric means between 324 and -12 are **-108** and **36**. The sequence is: \( 324, -108, 36, -12 \).

Would you like more details or have any questions? Here are some related topics:

1. How to determine the common ratio in geometric sequences.
2. What happens if the common ratio is a fraction?
3. How to find the sum of a finite geometric sequence.
4. How to solve geometric progression problems with unknown terms.
5. What are the implications of a negative common ratio in a sequence?

**Tip:** The formula \( a_n = a_1 \cdot r^{n-1} \) is powerful for quickly finding any term in a geometric sequence if you know the first term and the common ratio!

Learn how to solve a geometric sequence problem using the formula an = a1 * r^(n-1). This problem involves finding specific terms in a sequence with a given first term and common ratio. Explore step-by-step solution and related mathematical concepts.

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