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!