if x , 2y and 3z are in AP where the distinct numbers x, yand z are in gp. Then the common ratio of the GP is

Options

A. 3
B. `1/3`
C. 2
D. `1/2`

Explanation

To solve the problem, we need to find the common ratio \( r \) of the geometric progression (GP) given that \( x, 2y, \) and \( 3z \) are in arithmetic progression (AP) and \( x, y, z \) are distinct numbers in GP.

### Step-by-Step Solution:

1. **Understanding the Condition of AP**:
Since \( x, 2y, \) and \( 3z \) are in AP, we can use the property of AP which states that the middle term is the average of the other two terms. Therefore, we have:
\[ 2y = \frac{x + 3z}{2} \]
Multiplying both sides by 2 gives:
\[ 4y = x + 3z \tag{1} \]

2. **Expressing \( y \) and \( z \) in terms of \( x \) and \( r \)**:
Given that \( x, y, z \) are in GP, we can express \( y \) and \( z \) in terms of \( x \) and the common ratio \( r \):
\[ y = xr \quad \text{and} \quad z = xr^2 \]

3. **Substituting \( y \) and \( z \) into Equation (1)**:
Substitute \( y \) and \( z \) from the previous step into equation (1):
\[ 4(xr) = x + 3(xr^2) \]
Simplifying this gives:
\[ 4xr = x + 3xr^2 \]

4. **Rearranging the Equation**:
Rearranging the equation leads to:
\[ 4xr - 3xr^2 - x = 0 \]
Factoring out \( x \) (since \( x \neq 0 \)):
\[ x(4r - 3r^2 - 1) = 0 \]
This gives us the quadratic equation:
\[ 3r^2 - 4r + 1 = 0 \tag{2} \]

5. **Solving the Quadratic Equation**:
We can solve the quadratic equation (2) using the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 3, b = -4, c = 1 \):
\[ r = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 3 \cdot 1}}{2 \cdot 3} \]
\[ r = \frac{4 \pm \sqrt{16 - 12}}{6} \]
\[ r = \frac{4 \pm \sqrt{4}}{6} \]
\[ r = \frac{4 \pm 2}{6} \]
This gives us two possible values for \( r \):
\[ r = \frac{6}{6} = 1 \quad \text{and} \quad r = \frac{2}{6} = \frac{1}{3} \]

6. **Determining the Valid Common Ratio**:
Since \( x, y, z \) are distinct numbers in GP, \( r \) cannot be equal to 1 (as this would imply \( x = y = z \)). Thus, the only valid solution is:
\[ r = \frac{1}{3} \]

### Final Answer:
The common ratio of the GP is \( \frac{1}{3} \).