If `(x + y)/(2), y ,(y + z)/(2)` are in H.P., then x, y, z are in

Options

A. H.P.
B. G.P.
C. A.P.
D. none of these

Answer

To determine the relationship among \(x\), \(y\), and \(z\) given that \(\frac{x + y}{2}\), \(y\), and \(\frac{y + z}{2}\) are in Harmonic Progression (H.P.), we can follow these steps:

### Step 1: Understand the condition for H.P.
For three numbers \(A\), \(B\), and \(C\) to be in H.P., the condition is given by:
\[ \frac{2}{B} = \frac{1}{A} + \frac{1}{C} \]
Here, we will let:
- \(A = \frac{x + y}{2}\)
- \(B = y\)
- \(C = \frac{y + z}{2}\)

### Step 2: Substitute the values into the H.P. condition
Substituting the values of \(A\), \(B\), and \(C\) into the H.P. condition:
\[ \frac{2}{y} = \frac{1}{\frac{x + y}{2}} + \frac{1}{\frac{y + z}{2}} \]

### Step 3: Simplify the equation
Rewriting the right-hand side:
\[ \frac{2}{y} = \frac{2}{x + y} + \frac{2}{y + z} \]
Now, we can eliminate the denominators by multiplying through by \(y(x + y)(y + z)\):
\[ 2(x + y)(y + z) = 2y(y + z) + 2y(x + y) \]

### Step 4: Expand and simplify
Expanding both sides:
\[ 2(xy + xz + y^2 + yz) = 2y^2 + 2yz + 2xy + 2y^2 \]
This simplifies to:
\[ 2xy + 2xz + 2y^2 + 2yz = 2xy + 4y^2 + 2yz \]

### Step 5: Rearranging the equation
Now, we can rearrange the equation:
\[ 2xz + 2y^2 = 4y^2 \]
This simplifies to:
\[ 2xz = 2y^2 \]
Dividing both sides by 2 gives:
\[ xz = y^2 \]

### Step 6: Conclude the relationship
The equation \(xz = y^2\) shows that:
\[ \frac{y^2}{xz} = 1 \]
This implies that \(x\), \(y\), and \(z\) are in Geometric Progression (G.P.).

### Final Answer
Thus, we conclude that \(x\), \(y\), and \(z\) are in G.P.

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