If `x, 1, z` are in AP and `x, 2, z` are in GP, then `x, 4, z` will be in

Options

A. AP
B. GP
C. HP
D. None of these

Answer

To solve the problem step by step, we need to analyze the given information about the sequences.

### Step 1: Understand the conditions for AP
Given that \( x, 1, z \) are in Arithmetic Progression (AP), we know that the middle term is the average of the other two terms. Therefore, we can write the equation:

\[ 1 = \frac{x + z}{2} \]

### Step 2: Rearranging the AP equation
From the equation above, we can rearrange it to find a relationship between \( x \) and \( z \):

\[ x + z = 2 \quad \text{(Equation 1)} \]

### Step 3: Understand the conditions for GP
Next, we know that \( x, 2, z \) are in Geometric Progression (GP). For GP, the middle term is the geometric mean of the other two terms. Thus, we have:

\[ 2 = \sqrt{x \cdot z} \]

### Step 4: Squaring the GP equation
To eliminate the square root, we square both sides:

\[ 4 = xz \quad \text{(Equation 2)} \]

### Step 5: Solve the equations simultaneously
Now we have two equations:
1. \( x + z = 2 \)
2. \( xz = 4 \)

From Equation 1, we can express \( z \) in terms of \( x \):

\[ z = 2 - x \]

Substituting this expression for \( z \) into Equation 2 gives:

\[ x(2 - x) = 4 \]

### Step 6: Expand and rearrange the equation
Expanding the left side, we get:

\[ 2x - x^2 = 4 \]

Rearranging this gives us a standard quadratic equation:

\[ x^2 - 2x + 4 = 0 \]

### Step 7: Solve the quadratic equation
To solve the quadratic equation, we can use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 1, b = -2, c = 4 \):

\[ x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} \]
\[ x = \frac{2 \pm \sqrt{4 - 16}}{2} \]
\[ x = \frac{2 \pm \sqrt{-12}}{2} \]
\[ x = \frac{2 \pm 2i\sqrt{3}}{2} \]
\[ x = 1 \pm i\sqrt{3} \]

### Step 8: Find \( z \)
Using \( z = 2 - x \):

If \( x = 1 + i\sqrt{3} \), then:
\[ z = 2 - (1 + i\sqrt{3}) = 1 - i\sqrt{3} \]

If \( x = 1 - i\sqrt{3} \), then:
\[ z = 2 - (1 - i\sqrt{3}) = 1 + i\sqrt{3} \]

### Step 9: Determine the progression of \( x, 4, z \)
To check if \( x, 4, z \) are in Harmonic Progression (HP), we need to verify if:

\[ \frac{1}{4} = \frac{2}{x + z} \]

Since \( x + z = 2 \) from Equation 1, we have:

\[ \frac{1}{4} = \frac{2}{2} = 1 \]

This does not hold true. Therefore, we conclude that \( x, 4, z \) are not in HP.

### Conclusion
Thus, \( x, 4, z \) will be in Harmonic Progression (HP).

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