If a,b, and c are in G.P then a+b,2b and b+ c are in

Options

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

Explanation

To solve the problem, we need to determine the relationship between the terms \( A + B \), \( 2B \), and \( B + C \) given that \( A, B, C \) are in a geometric progression (G.P.).

### Step-by-Step Solution:

1. **Understanding the Geometric Progression (G.P.)**:
- If \( A, B, C \) are in G.P., then the ratio of consecutive terms is constant. This means:
\[ \frac{B}{A} = \frac{C}{B} \]
- We can express \( C \) in terms of \( A \) and \( B \):
\[ C = \frac{B^2}{A} \]

2. **Finding the New Terms**:
- We need to evaluate the terms \( A + B \), \( 2B \), and \( B + C \):
- \( A + B \)
- \( 2B \)
- \( B + C = B + \frac{B^2}{A} = B \left(1 + \frac{B}{A}\right) \)

3. **Substituting Values**:
- Let's assume \( A = 1 \), \( B = 2 \), and \( C = 4 \) (these values are in G.P. since \( 2^2 = 1 \times 4 \)):
- Calculate:
- \( A + B = 1 + 2 = 3 \)
- \( 2B = 2 \times 2 = 4 \)
- \( B + C = 2 + 4 = 6 \)

4. **Checking for Arithmetic Progression (A.P.)**:
- For \( A + B, 2B, B + C \) to be in A.P., the condition is:
\[ 2B - (A + B) = (B + C) - 2B \]
- Substituting the values:
\[ 4 - 3 = 6 - 4 \]
\[ 1 \neq 2 \quad \text{(not A.P.)} \]

5. **Checking for Geometric Progression (G.P.)**:
- For \( A + B, 2B, B + C \) to be in G.P., the condition is:
\[ \frac{2B}{A + B} = \frac{B + C}{2B} \]
- Substituting the values:
\[ \frac{4}{3} \neq \frac{6}{4} \quad \text{(not G.P.)} \]

6. **Checking for Harmonic Progression (H.P.)**:
- For \( A + B, 2B, B + C \) to be in H.P., the condition is:
\[ \frac{1}{A + B} + \frac{1}{B + C} = \frac{2}{2B} \]
- Substituting the values:
\[ \frac{1}{3} + \frac{1}{6} = \frac{2}{4} \]
\[ \frac{1}{3} + \frac{1}{6} = \frac{2}{4} \Rightarrow \frac{2}{6} = \frac{1}{3} \quad \text{(H.P. verified)} \]

### Conclusion:
Since \( A + B, 2B, B + C \) satisfy the condition for harmonic progression, the answer is:
\[ \text{They are in H.P.} \]