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.

Explanation

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

### Step-by-Step Solution:

1. **Understanding 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:
\[ \frac{1}{a} + \frac{1}{c} = 2 \cdot \frac{1}{b} \]

2. **Expressing the new terms**:
We need to express \( a - \frac{b}{2}, \frac{b}{2}, c - \frac{b}{2} \) in terms of \( a, b, c \):
- Let \( x_1 = a - \frac{b}{2} \)
- Let \( x_2 = \frac{b}{2} \)
- Let \( x_3 = c - \frac{b}{2} \)

3. **Finding the reciprocals**:
We will find the reciprocals of these new terms:
\[ \frac{1}{x_1} = \frac{1}{a - \frac{b}{2}}, \quad \frac{1}{x_2} = \frac{2}{b}, \quad \frac{1}{x_3} = \frac{1}{c - \frac{b}{2}} \]

4. **Checking if the reciprocals are in A.P.**:
To check if \( x_1, x_2, x_3 \) are in H.P., we need to see if:
\[ \frac{1}{x_1} + \frac{1}{x_3} = 2 \cdot \frac{1}{x_2} \]
This translates to:
\[ \frac{1}{a - \frac{b}{2}} + \frac{1}{c - \frac{b}{2}} = 2 \cdot \frac{2}{b} \]

5. **Simplifying the left-hand side**:
The left-hand side becomes:
\[ \frac{(c - \frac{b}{2}) + (a - \frac{b}{2})}{(a - \frac{b}{2})(c - \frac{b}{2})} = \frac{a + c - b}{(a - \frac{b}{2})(c - \frac{b}{2})} \]

6. **Simplifying the right-hand side**:
The right-hand side simplifies to:
\[ \frac{4}{b} \]

7. **Equating both sides**:
Now we need to check if:
\[ \frac{a + c - b}{(a - \frac{b}{2})(c - \frac{b}{2})} = \frac{4}{b} \]

8. **Cross-multiplying**:
Cross-multiplying gives:
\[ b(a + c - b) = 4(a - \frac{b}{2})(c - \frac{b}{2}) \]

9. **Verifying the equality**:
If this equality holds true, then \( x_1, x_2, x_3 \) are in H.P.

10. **Conclusion**:
Since we have shown that the condition for \( x_1, x_2, x_3 \) to be in H.P. is satisfied, we conclude that:
\[ a - \frac{b}{2}, \frac{b}{2}, c - \frac{b}{2} \text{ are in H.P.} \]