If three numbers are in H.P., then the numbers obtained by subtracting half of the middle number from each of them are in

Options

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

Answer

To solve the problem step by step, we need to analyze the situation where three numbers are in Harmonic Progression (H.P.) and then determine the nature of the numbers obtained by subtracting half of the middle number from each of them.

### Step-by-Step Solution:

1. **Understanding H.P.**:
- If three numbers \( A, B, C \) are in H.P., it means that their reciprocals \( \frac{1}{A}, \frac{1}{B}, \frac{1}{C} \) are in Arithmetic Progression (A.P.).
- Therefore, we have the relation:
\[ \frac{1}{A} + \frac{1}{C} = 2 \cdot \frac{1}{B} \]

2. **Rearranging the H.P. condition**:
- From the H.P. condition, we can rearrange it to:
\[ \frac{A + C}{AC} = \frac{2}{B} \]
- This implies:
\[ B = \frac{2AC}{A + C} \]

3. **Subtracting half of the middle number**:
- We need to find the numbers obtained by subtracting half of the middle number \( B \) from each of the numbers \( A, B, C \).
- Half of \( B \) is \( \frac{B}{2} \).
- The new numbers will be:
\[ A - \frac{B}{2}, \quad B - \frac{B}{2}, \quad C - \frac{B}{2} \]
- This simplifies to:
\[ A - \frac{B}{2}, \quad \frac{B}{2}, \quad C - \frac{B}{2} \]

4. **Finding the product of the first and last terms**:
- We will now check if the new numbers are in Geometric Progression (G.P.) by examining the product of the first and last terms:
\[ (A - \frac{B}{2})(C - \frac{B}{2}) \]

5. **Expanding the product**:
- Expanding this gives:
\[ AC - \frac{AB}{2} - \frac{BC}{2} + \frac{B^2}{4} \]

6. **Substituting \( B \)**:
- We substitute \( B = \frac{2AC}{A + C} \) into the expression:
\[ AC - \frac{1}{2} \cdot \frac{2AC}{A + C} \cdot A - \frac{1}{2} \cdot \frac{2AC}{A + C} \cdot C + \frac{1}{4} \cdot \left(\frac{2AC}{A + C}\right)^2 \]

7. **Simplifying**:
- After simplification, we find that the product simplifies to:
\[ \left(\frac{B}{2}\right)^2 \]
- This shows that the new numbers are in G.P. because the product of the first and last terms equals the square of the middle term.

### Conclusion:
Thus, the numbers obtained by subtracting half of the middle number from each of the three numbers in H.P. are in **Geometric Progression (G.P.)**.