Insert 3 geometric means between 16 and 256.

Answer

To insert 3 geometric means between 16 and 256, we can follow these steps:

### Step 1: Understand the structure of the geometric progression (GP)
In a GP, if we have the first term \( a \) and the common ratio \( r \), the terms can be expressed as:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)
- Fourth term: \( ar^3 \)
- Fifth term: \( ar^4 \)

### Step 2: Identify the terms
Here, we want to insert three geometric means between 16 and 256. So, we can denote:
- First term \( a = 16 \)
- Fifth term \( ar^4 = 256 \)

### Step 3: Set up the equation
From the above, we can set up the equation:
\[ ar^4 = 256 \]
Substituting \( a = 16 \):
\[ 16r^4 = 256 \]

### Step 4: Solve for \( r^4 \)
To find \( r^4 \), divide both sides by 16:
\[ r^4 = \frac{256}{16} = 16 \]

### Step 5: Solve for \( r \)
Now, take the fourth root of both sides:
\[ r = \sqrt[4]{16} \]
Since \( 16 = 2^4 \), we have:
\[ r = 2 \]

### Step 6: Calculate the geometric means
Now that we have \( r \), we can find the three geometric means:
- First geometric mean \( a_1 = ar = 16 \times 2 = 32 \)
- Second geometric mean \( a_2 = ar^2 = 16 \times 2^2 = 16 \times 4 = 64 \)
- Third geometric mean \( a_3 = ar^3 = 16 \times 2^3 = 16 \times 8 = 128 \)

### Step 7: Write the final answer
Thus, the three geometric means between 16 and 256 are:
\[ 32, 64, 128 \]

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