Five geometric means between 486 and 2/3 are………………

Answer

To find the five geometric means between 486 and \( \frac{2}{3} \), we can follow these steps:

### Step 1: Identify the terms
Let the first term \( A = 486 \) and the seventh term \( T_7 = \frac{2}{3} \). We need to find the five geometric means, which will be the second to sixth terms \( G_1, G_2, G_3, G_4, G_5 \).

### Step 2: Use the formula for the nth term of a geometric progression
The nth term of a geometric progression can be expressed as:
\[ T_n = A \cdot R^{n-1} \]
where \( A \) is the first term and \( R \) is the common ratio.

### Step 3: Set up the equation for the seventh term
For the seventh term:
\[ T_7 = A \cdot R^{7-1} = A \cdot R^6 \]
Substituting the known values:
\[ \frac{2}{3} = 486 \cdot R^6 \]

### Step 4: Solve for \( R^6 \)
Rearranging the equation gives:
\[ R^6 = \frac{2/3}{486} \]
Calculating \( 486 \):
\[ 486 = \frac{1458}{3} \implies R^6 = \frac{2}{3} \cdot \frac{3}{1458} = \frac{2}{1458} \]
Simplifying \( \frac{2}{1458} \):
\[ R^6 = \frac{1}{729} \quad \text{(since } 1458 = 729 \cdot 2\text{)} \]

### Step 5: Find \( R \)
Taking the sixth root:
\[ R = \left( \frac{1}{729} \right)^{1/6} = \frac{1}{3} \]

### Step 6: Calculate the geometric means
Now, we can find the geometric means using the formula \( G_n = A \cdot R^{n-1} \).

1. **First geometric mean \( G_1 \)**:
\[ G_1 = A \cdot R^{1} = 486 \cdot \frac{1}{3} = 162 \]

2. **Second geometric mean \( G_2 \)**:
\[ G_2 = A \cdot R^{2} = 486 \cdot \left( \frac{1}{3} \right)^2 = 486 \cdot \frac{1}{9} = 54 \]

3. **Third geometric mean \( G_3 \)**:
\[ G_3 = A \cdot R^{3} = 486 \cdot \left( \frac{1}{3} \right)^3 = 486 \cdot \frac{1}{27} = 18 \]

4. **Fourth geometric mean \( G_4 \)**:
\[ G_4 = A \cdot R^{4} = 486 \cdot \left( \frac{1}{3} \right)^4 = 486 \cdot \frac{1}{81} = 6 \]

5. **Fifth geometric mean \( G_5 \)**:
\[ G_5 = A \cdot R^{5} = 486 \cdot \left( \frac{1}{3} \right)^5 = 486 \cdot \frac{1}{243} = 2 \]

### Final Answer
The five geometric means between 486 and \( \frac{2}{3} \) are:
\[ 162, 54, 18, 6, 2 \]