To find 4 geometric means between 3 and 48, we are looking to construct a geometric sequence where 3 is the first term, 48 is the last term, and there are 4 terms in between them. Let's denote the sequence as:

\[
3, \, a_1, \, a_2, \, a_3, \, a_4, \, 48
\]

### Step 1: Understanding the Geometric Sequence

In a geometric sequence, each term is obtained by multiplying the previous term by a common ratio \( r \). Thus, we can write:

\[
a_1 = 3r, \quad a_2 = 3r^2, \quad a_3 = 3r^3, \quad a_4 = 3r^4, \quad 48 = 3r^5
\]

### Step 2: Solving for the Common Ratio \( r \)

To find \( r \), we use the last term:

\[
3r^5 = 48
\]

Dividing both sides by 3:

\[
r^5 = 16
\]

Now, take the fifth root of both sides:

\[
r = 16^{1/5}
\]

### Step 3: Calculating the Common Ratio

Calculating \( 16^{1/5} \):

\[
r = 2
\]

### Step 4: Finding the Geometric Means

Now that we have \( r = 2 \), we can find the 4 geometric means:

\[
a_1 = 3r = 3 \times 2 = 6
\]
\[
a_2 = 3r^2 = 3 \times 4 = 12
\]
\[
a_3 = 3r^3 = 3 \times 8 = 24
\]
\[
a_4 = 3r^4 = 3 \times 16 = 48
\]

### Conclusion

The 4 geometric means between 3 and 48 are:

\[
6, \, 12, \, 24, \, 48
\]

Would you like more details or have any questions? Here are 5 related questions for further exploration:

1. How do you find geometric means in a different range, such as between 5 and 125?
2. What is the formula for finding the \( n \)-th term of a geometric sequence?
3. How does arithmetic progression differ from geometric progression?
4. How can we determine the sum of a geometric series?
5. What are the applications of geometric sequences in real-world scenarios?

**Tip:** When working with geometric sequences, always check if the terms grow by a constant ratio, confirming that the sequence is geometric.

Discover how to find 4 geometric means between 3 and 48. This solution explains the concept of geometric sequences, the formula for calculating geometric means, and provides a step-by-step guide. Ideal for students in Grades 9-10.

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