If a,b,c are in H.P. then `a-(b)/(2),(b)/(2),c-(b)/(2)` are in

Options

A. A.P
B. G.P
C. A.G.P
D. H.P

Answer

To determine the relationship between the terms \( a - \frac{b}{2}, \frac{b}{2}, c - \frac{b}{2} \) given that \( a, b, c \) are in Harmonic Progression (H.P.), we can follow these steps:

### Step 1: Understand the Condition of H.P.
If \( a, b, c \) are in H.P., then their reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) are in Arithmetic Progression (A.P.). This means:
\[ 2 \cdot \frac{1}{b} = \frac{1}{a} + \frac{1}{c} \]
From this, we can derive:
\[ \frac{2}{b} = \frac{1}{a} + \frac{1}{c} \]

### Step 2: Express \( b \) in terms of \( a \) and \( c \)
Rearranging the equation gives us:
\[ \frac{2ac}{b} = a + c \implies b = \frac{2ac}{a+c} \]

### Step 3: Rewrite the New Terms
Now, we need to analyze the terms \( a - \frac{b}{2}, \frac{b}{2}, c - \frac{b}{2} \):
- Let \( x = a - \frac{b}{2} \)
- Let \( y = \frac{b}{2} \)
- Let \( z = c - \frac{b}{2} \)

### Step 4: Check if \( x, y, z \) are in Geometric Progression (G.P.)
To check if \( x, y, z \) are in G.P., we need to verify if:
\[ y^2 = xz \]

Calculating \( x \) and \( z \):
- \( x = a - \frac{b}{2} = a - \frac{1}{2} \cdot \frac{2ac}{a+c} = a - \frac{ac}{a+c} = \frac{a(a+c) - ac}{a+c} = \frac{a^2}{a+c} \)
- \( z = c - \frac{b}{2} = c - \frac{ac}{a+c} = \frac{c(a+c) - ac}{a+c} = \frac{c^2}{a+c} \)

### Step 5: Calculate \( y^2 \)
Now, calculate \( y^2 \):
\[ y = \frac{b}{2} = \frac{1}{2} \cdot \frac{2ac}{a+c} = \frac{ac}{a+c} \]
Thus,
\[ y^2 = \left(\frac{ac}{a+c}\right)^2 \]

### Step 6: Check the G.P. Condition
Now we need to check if:
\[ \left(\frac{ac}{a+c}\right)^2 = \left(\frac{a^2}{a+c}\right)\left(\frac{c^2}{a+c}\right) \]
This simplifies to:
\[ \frac{a^2c^2}{(a+c)^2} = \frac{a^2c^2}{(a+c)^2} \]
This equality holds true, confirming that \( x, y, z \) are indeed in G.P.

### Conclusion
Thus, we conclude that if \( a, b, c \) are in H.P., then \( a - \frac{b}{2}, \frac{b}{2}, c - \frac{b}{2} \) are in G.P.

---