16 and 81	[3]  (find the indicated number of geometric mean between each pair of numbers)

To find the indicated number of geometric means between two numbers in a geometric sequence, you can follow these steps:

### Step 1: Identify the given values
- **First term \( a_1 \):** 16
- **Last term \( a_{n+1} \):** 81
- **Number of geometric means ( \( n \) ):** 3

### Step 2: Use the formula for the general term of a geometric sequence
The general formula for the \( k \)-th term of a geometric sequence is:
\[
a_k = a_1 \cdot r^{k-1}
\]
where:
- \( a_1 \) is the first term,
- \( r \) is the common ratio, and
- \( k \) is the term number.

### Step 3: Find the common ratio \( r \)
Since there are 3 geometric means, the sequence has a total of 5 terms (including the first and last term). So:
\[
81 = 16 \cdot r^4
\]
Now solve for \( r \):
\[
r^4 = \frac{81}{16}
\]
\[
r^4 = \left(\frac{9}{4}\right)^2
\]
\[
r = \sqrt[4]{\left(\frac{9}{4}\right)^2} = \sqrt{\frac{9}{4}} = \frac{3}{2}
\]

### Step 4: Find the geometric means
The sequence is:
\[
16, \, 16 \cdot \frac{3}{2}, \, 16 \cdot \left(\frac{3}{2}\right)^2, \, 16 \cdot \left(\frac{3}{2}\right)^3, \, 81
\]

Calculating the terms:
- **First Geometric Mean:** \( 16 \cdot \frac{3}{2} = 24 \)
- **Second Geometric Mean:** \( 16 \cdot \left(\frac{3}{2}\right)^2 = 36 \)
- **Third Geometric Mean:** \( 16 \cdot \left(\frac{3}{2}\right)^3 = 54 \)

### Answer
The 3 geometric means between 16 and 81 are **24, 36, and 54**.

Would you like more details or have any questions?

Here are 5 related questions to further your understanding:

1. How do you find the common ratio in a geometric sequence with different numbers of means?
2. What would the geometric sequence look like if the common ratio was negative?
3. How do you calculate the geometric means if the initial and final terms are negative?
4. How does the formula for geometric means differ from that for arithmetic means?
5. Can you calculate the geometric means if the sequence involves fractional numbers?

**Tip:** When dealing with sequences, double-check that the number of terms matches the given sequence type (arithmetic or geometric) to avoid errors.

16 and 81 [3] (find the indicated number of geometric mean between each pair of numbers)

Learn how to find the 3 geometric means between 16 and 81 using geometric sequences. This problem covers key mathematical concepts, including the formula for the general term of a geometric sequence and the calculation of the common ratio. Suitable for students in Grades 9-12.

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