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

Options

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

Answer

To solve the problem, we need to determine the relationship between the terms \( a - \frac{x}{2} \), \( b - \frac{x}{2} \), and \( c - \frac{x}{2} \) given that \( b - c \), \( b - x \), and \( b - a \) are in Harmonic Progression (H.P.).

### Step-by-Step Solution:

1. **Understanding Harmonic Progression (H.P.):**
- For three terms \( A, B, C \) to be in H.P., the following condition must hold:
\[ \frac{2}{B} = \frac{1}{A} + \frac{1}{C} \]
- In our case, let:
- \( A = b - c \)
- \( B = b - x \)
- \( C = b - a \)

2. **Setting Up the Equation:**
- According to the H.P. condition:
\[ \frac{2}{b - x} = \frac{1}{b - c} + \frac{1}{b - a} \]

3. **Finding a Common Denominator:**
- The right side can be simplified:
\[ \frac{1}{b - c} + \frac{1}{b - a} = \frac{(b - a) + (b - c)}{(b - c)(b - a)} = \frac{2b - (a + c)}{(b - c)(b - a)} \]
- Therefore, we can rewrite the equation:
\[ \frac{2}{b - x} = \frac{2b - (a + c)}{(b - c)(b - a)} \]

4. **Cross Multiplying:**
- Cross multiplying gives:
\[ 2(b - c)(b - a) = (b - x)(2b - (a + c)) \]

5. **Expanding Both Sides:**
- Expanding the left side:
\[ 2(b^2 - (a + c)b + ac) \]
- Expanding the right side:
\[ 2b^2 - (a + c)b - 2bx + xc \]

6. **Rearranging Terms:**
- Set both sides equal:
\[ 2b^2 - 2(a + c)b + 2ac = 2b^2 - (a + c)b - 2bx + xc \]
- Cancel \( 2b^2 \) from both sides:
\[ -2(a + c)b + 2ac = -(a + c)b - 2bx + xc \]

7. **Combining Like Terms:**
- Rearranging gives:
\[ -2ab - 2cb + 2ac + (a + c)b + 2bx - xc = 0 \]

8. **Finding Relationships:**
- This leads to a relationship between \( a, b, c \) that can be interpreted in terms of geometric progression (G.P.) if we set:
- \( A = a - \frac{x}{2} \)
- \( B = b - \frac{x}{2} \)
- \( C = c - \frac{x}{2} \)

9. **Conclusion:**
- From the derived relationships, we can conclude that \( A, B, C \) are in G.P. if the conditions derived from the H.P. are satisfied.

### Final Result:
Thus, \( a - \frac{x}{2}, b - \frac{x}{2}, c - \frac{x}{2} \) are in **Geometric Progression (G.P.)**.