To find three geometric means between 16 and 81, we'll use the geometric sequence formula.

The formula for the \(n\)-th term of a geometric sequence is:

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

Where:
- \(a_1 = 16\) (the first term)
- \(a_5 = 81\) (the fifth term, since there are 3 geometric means between them, making the total sequence length 5)
- \(r\) is the common ratio, which we need to find.

We know:

\[
a_5 = a_1 \cdot r^4
\]

Substitute the known values:

\[
81 = 16 \cdot r^4
\]

Now, solve for \(r\):

\[
r^4 = \frac{81}{16} = 5.0625
\]

Take the fourth root of both sides:

\[
r = \sqrt[4]{5.0625} \approx 1.5
\]

Now that we have the common ratio \(r = 1.5\), we can find the three geometric means:

1. The second term \(a_2 = a_1 \cdot r = 16 \cdot 1.5 = 24\)
2. The third term \(a_3 = a_2 \cdot r = 24 \cdot 1.5 = 36\)
3. The fourth term \(a_4 = a_3 \cdot r = 36 \cdot 1.5 = 54\)

Thus, the three geometric means between 16 and 81 are:

\[
\boxed{24}, \boxed{36}, \boxed{54}
\]

Would you like more details or have any questions?

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Here are five related questions to expand on this topic:

1. How do you find the geometric mean of two numbers?
2. What is the difference between arithmetic and geometric sequences?
3. Can you derive the formula for the \(n\)-th term of a geometric sequence?
4. How do you solve a geometric sequence problem with more than three means?
5. What happens if one or both of the terms are negative?

**Tip:** In a geometric sequence, each term is a constant multiple (ratio) of the previous term!

Find 3 geometric means between 16 and 81.

Learn how to find three geometric means between 16 and 81 using the geometric sequence formula. This step-by-step guide covers key concepts, including calculating the common ratio and applying the geometric sequence formula. Ideal for students in Grades 8-10.

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