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 of the geometric progression (GP) given that \( x \), \( 2y \), and \( 3z \) are in arithmetic progression (AP) and that \( x \), \( y \), and \( z \) are distinct numbers in GP.

### Step-by-Step Solution:

1. **Understanding the AP Condition**:
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. Thus, we have:
\[ 2(2y) = x + 3z \]
Simplifying this gives:
\[ 4y = x + 3z \quad \text{(Equation 1)} \]

2. **Understanding the GP Condition**:
Since \( x \), \( y \), and \( z \) 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 GP expressions into Equation 1:
\[ 4(xr) = x + 3(xr^2) \]
This simplifies to:
\[ 4xr = x + 3xr^2 \]

4. **Rearranging the Equation**:
We can rearrange the equation to isolate terms involving \( x \):
\[ 4xr - 3xr^2 = x \]
Factor out \( x \) (assuming \( x \neq 0 \)):
\[ x(4r - 3r^2 - 1) = 0 \]
Since \( x \neq 0 \), we can divide by \( x \):
\[ 4r - 3r^2 - 1 = 0 \]

5. **Rearranging to Form a Quadratic Equation**:
Rearranging gives us:
\[ 3r^2 - 4r + 1 = 0 \]

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

7. **Considering the Distinctness Condition**:
Since \( x \), \( y \), and \( z \) are distinct, we cannot have \( r = 1 \) (which would imply \( x = y = z \)). Therefore, we discard \( r = 1 \).

8. **Final Result**:
The only valid solution is:
\[ r = \frac{1}{3} \]

Thus, the common ratio of the GP is \( \frac{1}{3} \).