Find 5 geometric means between 1 and 27.

Answer

To find 5 geometric means between 1 and 27, we can follow these steps:

### Step 1: Understand the Problem
We need to find 5 geometric means between the numbers 1 and 27. This means we will have a total of 7 terms in the geometric sequence: the first term (1), five geometric means, and the last term (27).

### Step 2: Define the Terms
Let the first term \( a_1 = 1 \) and the seventh term \( a_7 = 27 \). The common ratio will be denoted as \( r \).

### Step 3: Use the Formula for the nth Term of a Geometric Sequence
The formula for the nth term of a geometric sequence is given by:
\[ a_n = a \cdot r^{(n-1)} \]
For our case, the seventh term can be expressed as:
\[ a_7 = a_1 \cdot r^{(7-1)} = 1 \cdot r^6 \]
Thus, we have:
\[ r^6 = 27 \]

### Step 4: Solve for the Common Ratio \( r \)
We can rewrite 27 as \( 3^3 \):
\[ r^6 = 3^3 \]
Taking the sixth root of both sides, we get:
\[ r = 27^{1/6} = (3^3)^{1/6} = 3^{3/6} = 3^{1/2} = \sqrt{3} \]

### Step 5: Find the Geometric Means
Now that we have the common ratio \( r = \sqrt{3} \), we can find the five geometric means:
1. \( a_2 = a_1 \cdot r = 1 \cdot \sqrt{3} = \sqrt{3} \)
2. \( a_3 = a_2 \cdot r = \sqrt{3} \cdot \sqrt{3} = 3 \)
3. \( a_4 = a_3 \cdot r = 3 \cdot \sqrt{3} = 3\sqrt{3} \)
4. \( a_5 = a_4 \cdot r = 3\sqrt{3} \cdot \sqrt{3} = 9 \)
5. \( a_6 = a_5 \cdot r = 9 \cdot \sqrt{3} = 9\sqrt{3} \)

### Step 6: List the Geometric Means
The five geometric means between 1 and 27 are:
- \( \sqrt{3} \)
- \( 3 \)
- \( 3\sqrt{3} \)
- \( 9 \)
- \( 9\sqrt{3} \)

### Final Answer
The five geometric means between 1 and 27 are:
\[ \sqrt{3}, 3, 3\sqrt{3}, 9, 9\sqrt{3} \]

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