Find 3 geometric means between 10 and 160.

Explanation

To find 3 geometric means between 10 and 160, we can follow these steps:

### Step 1: Set up the geometric sequence
Let the three geometric means be \( g_1, g_2, g_3 \). The sequence will be:
\[ 10, g_1, g_2, g_3, 160 \]

### Step 2: Express the terms in terms of the first term and common ratio
Let \( A = 10 \) (the first term) and \( R \) be the common ratio. The terms can be expressed as follows:
- \( g_1 = A \cdot R = 10R \)
- \( g_2 = A \cdot R^2 = 10R^2 \)
- \( g_3 = A \cdot R^3 = 10R^3 \)
- The last term is \( g_4 = A \cdot R^4 = 10R^4 = 160 \)

### Step 3: Set up the equation for the last term
From the last term, we have:
\[ 10R^4 = 160 \]

### Step 4: Solve for \( R^4 \)
Dividing both sides by 10:
\[ R^4 = \frac{160}{10} = 16 \]

### Step 5: Find \( R \)
Taking the fourth root of both sides:
\[ R = \sqrt[4]{16} = \sqrt[4]{2^4} = 2 \quad \text{or} \quad R = -2 \]

### Step 6: Calculate the geometric means for \( R = 2 \)
Now, substituting \( R = 2 \):
- \( g_1 = 10R = 10 \cdot 2 = 20 \)
- \( g_2 = 10R^2 = 10 \cdot 2^2 = 10 \cdot 4 = 40 \)
- \( g_3 = 10R^3 = 10 \cdot 2^3 = 10 \cdot 8 = 80 \)

### Step 7: Calculate the geometric means for \( R = -2 \)
Now, substituting \( R = -2 \):
- \( g_1 = 10R = 10 \cdot (-2) = -20 \)
- \( g_2 = 10R^2 = 10 \cdot (-2)^2 = 10 \cdot 4 = 40 \)
- \( g_3 = 10R^3 = 10 \cdot (-2)^3 = 10 \cdot (-8) = -80 \)

### Conclusion
The three geometric means between 10 and 160 are:
1. For \( R = 2 \): \( 20, 40, 80 \)
2. For \( R = -2 \): \( -20, 40, -80 \)