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.