If `b-c,2b-lambda,b-a " are in HP, then " a-(lambda)/(2),b-(lambda)/(2),c-(lambda)/(2)` are is

Options

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

Explanation

To solve the problem, we need to determine the type of progression for the terms \( a - \frac{\lambda}{2}, b - \frac{\lambda}{2}, c - \frac{\lambda}{2} \) given that \( b - c, 2b - \lambda, b - a \) are in Harmonic Progression (HP).

### Step-by-Step Solution:

1. **Understanding Harmonic Progression (HP)**:
- If three terms \( x, y, z \) are in HP, then the reciprocals \( \frac{1}{x}, \frac{1}{y}, \frac{1}{z} \) are in Arithmetic Progression (AP).
- Thus, for our terms \( b - c, 2b - \lambda, b - a \) to be in HP, we can write:
\[ \frac{1}{b - c}, \frac{1}{2b - \lambda}, \frac{1}{b - a} \]
must be in AP.

2. **Setting Up the Equation**:
- The condition for three numbers \( x, y, z \) to be in AP is:
\[ 2y = x + z \]
Applying this to our terms:
\[ 2(2b - \lambda) = (b - c) + (b - a) \]
Simplifying this gives:
\[ 4b - 2\lambda = 2b - (c + a) \]

3. **Rearranging the Equation**:
- Rearranging the above equation:
\[ 4b - 2b + c + a = 2\lambda \]
This simplifies to:
\[ 2b + a + c = 2\lambda \]
Thus, we can express \( \lambda \) as:
\[ \lambda = \frac{2b + a + c}{2} \]

4. **Substituting Back**:
- Now we substitute \( \lambda \) into the terms \( a - \frac{\lambda}{2}, b - \frac{\lambda}{2}, c - \frac{\lambda}{2} \):
\[ a - \frac{2b + a + c}{4}, \quad b - \frac{2b + a + c}{4}, \quad c - \frac{2b + a + c}{4} \]

5. **Simplifying Each Term**:
- For \( a - \frac{2b + a + c}{4} \):
\[ = \frac{4a - (2b + a + c)}{4} = \frac{3a - 2b - c}{4} \]
- For \( b - \frac{2b + a + c}{4} \):
\[ = \frac{4b - (2b + a + c)}{4} = \frac{2b - a - c}{4} \]
- For \( c - \frac{2b + a + c}{4} \):
\[ = \frac{4c - (2b + a + c)}{4} = \frac{3c - 2b - a}{4} \]

6. **Final Form**:
- Thus, we have the three terms:
\[ \frac{3a - 2b - c}{4}, \quad \frac{2b - a - c}{4}, \quad \frac{3c - 2b - a}{4} \]
- Now, we check if these terms form a geometric progression (GP).

7. **Condition for GP**:
- For three terms \( x, y, z \) to be in GP, the condition is:
\[ y^2 = xz \]
- Substituting our terms into this condition will confirm if they are in GP.

### Conclusion:
By following the above steps, we conclude that \( a - \frac{\lambda}{2}, b - \frac{\lambda}{2}, c - \frac{\lambda}{2} \) are in **Geometric Progression (GP)**.