Insert 3 geometric means between 3 and 48.

Answer

To insert 3 geometric means between the numbers 3 and 48, we can follow these steps:

### Step 1: Understand the structure of a geometric progression (GP)
In a geometric progression, each term after the first is found by multiplying the previous term by a constant called the common ratio (r). If we denote the first term as \( a \) and the number of terms as \( n \), the \( n \)-th term can be expressed as:
\[ T_n = a \cdot r^{n-1} \]

### Step 2: Identify the known values
Here, we have:
- First term \( a = 3 \)
- Fifth term \( T_5 = 48 \)

### Step 3: Set up the equation for the fifth term
Using the formula for the \( n \)-th term:
\[ T_5 = a \cdot r^{5-1} = a \cdot r^4 \]
Substituting the known values:
\[ 48 = 3 \cdot r^4 \]

### Step 4: Solve for the common ratio \( r \)
To find \( r^4 \), divide both sides by 3:
\[ r^4 = \frac{48}{3} = 16 \]
Now, take the fourth root of both sides to find \( r \):
\[ r = 2 \]

### Step 5: Calculate the geometric means
Now that we have \( r = 2 \), we can find the three geometric means (G1, G2, G3) as follows:

1. **First geometric mean (G1)**:
\[ G1 = a \cdot r^1 = 3 \cdot 2^1 = 3 \cdot 2 = 6 \]

2. **Second geometric mean (G2)**:
\[ G2 = a \cdot r^2 = 3 \cdot 2^2 = 3 \cdot 4 = 12 \]

3. **Third geometric mean (G3)**:
\[ G3 = a \cdot r^3 = 3 \cdot 2^3 = 3 \cdot 8 = 24 \]

### Step 6: List the geometric means
The three geometric means between 3 and 48 are:
- G1 = 6
- G2 = 12
- G3 = 24

### Final Answer
The three geometric means between 3 and 48 are **6, 12, and 24**.

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