Insert three geometric means between 1 and 256.

Answer

To insert three geometric means between 1 and 256, we can follow these steps:

### Step 1: Understand the sequence
We need to insert three geometric means between the numbers 1 and 256. This means we will have the sequence:
\[ 1, g_1, g_2, g_3, 256 \]

### Step 2: Identify the first term and the last term
Here, the first term \( a = 1 \) and the last term \( a_5 = 256 \).

### Step 3: Determine the number of terms
Since we are inserting three geometric means, the total number of terms in the sequence is 5.

### Step 4: Use the formula for the nth term of a geometric sequence
The nth term of a geometric sequence can be expressed as:
\[ a_n = a \cdot r^{(n-1)} \]
where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

### Step 5: Set up the equation for the last term
For the last term, we have:
\[ a_5 = a \cdot r^{(5-1)} \]
Substituting the known values:
\[ 256 = 1 \cdot r^4 \]
This simplifies to:
\[ r^4 = 256 \]

### Step 6: Solve for the common ratio \( r \)
To find \( r \), we take the fourth root of both sides:
\[ r = 256^{1/4} \]

### Step 7: Simplify \( r \)
We can express 256 as \( 4^4 \):
\[ r = (4^4)^{1/4} = 4 \]

### Step 8: Find the geometric means
Now that we have \( r = 4 \), we can find the three geometric means:
- The first geometric mean \( g_1 \):
\[ g_1 = a \cdot r = 1 \cdot 4 = 4 \]
- The second geometric mean \( g_2 \):
\[ g_2 = a \cdot r^2 = 1 \cdot 4^2 = 16 \]
- The third geometric mean \( g_3 \):
\[ g_3 = a \cdot r^3 = 1 \cdot 4^3 = 64 \]

### Step 9: Write the complete sequence
Thus, the complete sequence with the three geometric means inserted is:
\[ 1, 4, 16, 64, 256 \]

### Final Answer
The three geometric means between 1 and 256 are:
\[ 4, 16, 64 \]

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