If x, 3/2, z are in AP, x, 3, z are in GP, then which of the following will be in HP.

Options

A. A)x,6,z
B. B)x,4,z
C. C)x,2,z
D. D)x,1,z

Answer

To solve the problem step by step, we need to use the properties of Arithmetic Progression (AP), Geometric Progression (GP), and Harmonic Progression (HP).

### Step 1: Understanding the given conditions
We are given two sets of numbers:
1. \( x, \frac{3}{2}, z \) are in AP.
2. \( x, 3, z \) are in GP.

### Step 2: Using the property of AP
For numbers to be in AP, the middle term is the average of the other two terms. Thus, we can write:
\[ \frac{3}{2} = \frac{x + z}{2} \]
Multiplying both sides by 2 gives:
\[ 3 = x + z \quad \text{(Equation 1)} \]

### Step 3: Using the property of GP
For numbers to be in GP, the square of the middle term is equal to the product of the other two terms. Thus, we can write:
\[ 3^2 = x \cdot z \]
This simplifies to:
\[ 9 = xz \quad \text{(Equation 2)} \]

### Step 4: Solving the equations
Now we have two equations:
1. \( x + z = 3 \)
2. \( xz = 9 \)

From Equation 1, we can express \( z \) in terms of \( x \):
\[ z = 3 - x \]

Substituting this into Equation 2:
\[ x(3 - x) = 9 \]
Expanding this gives:
\[ 3x - x^2 = 9 \]
Rearranging leads to:
\[ x^2 - 3x + 9 = 0 \]

### Step 5: Finding the roots of the quadratic equation
Now we will use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -3, c = 9 \):
\[ x = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 9}}{2 \cdot 1} \]
Calculating the discriminant:
\[ (-3)^2 - 4 \cdot 1 \cdot 9 = 9 - 36 = -27 \]
Since the discriminant is negative, \( x \) and \( z \) are complex numbers.

### Step 6: Finding the Harmonic Mean
We know the relationship between Arithmetic Mean (A), Geometric Mean (G), and Harmonic Mean (H):
\[ A \cdot H = G^2 \]
From our previous steps:
- The Arithmetic Mean \( A = \frac{3}{2} \)
- The Geometric Mean \( G = 3 \)

Substituting these values into the equation:
\[ \frac{3}{2} \cdot H = 3^2 \]
This simplifies to:
\[ \frac{3}{2} \cdot H = 9 \]
Multiplying both sides by \( \frac{2}{3} \):
\[ H = 9 \cdot \frac{2}{3} = 6 \]

### Step 7: Forming the Harmonic Progression
The harmonic mean \( H = 6 \) means that for \( x, z \) to be in HP, we need:
\[ x, 6, z \]
Thus, \( x, 6, z \) will be in Harmonic Progression.

### Conclusion
The correct answer is that \( x, 6, z \) will be in HP.