If a, b, c are in A.P. as well as in G.P. then correct statement is :

Options

A. `a=b=c`
B. `anebnec`
C. `a=bnec`
D. None of these

Explanation

To solve the problem, we need to analyze the conditions given: \(a\), \(b\), and \(c\) are in both Arithmetic Progression (A.P.) and Geometric Progression (G.P.).

### Step-by-Step Solution:

1. **Understanding A.P. Condition**:
- If \(a\), \(b\), and \(c\) are in A.P., then the middle term \(b\) can be expressed as:
\[ b = \frac{a + c}{2} \]

2. **Understanding G.P. Condition**:
- If \(a\), \(b\), and \(c\) are in G.P., then the square of the middle term \(b\) is equal to the product of the other two terms:
\[ b^2 = ac \]

3. **Substituting A.P. into G.P.**:
- From the A.P. condition, we substitute \(b\) into the G.P. condition:
\[ \left(\frac{a + c}{2}\right)^2 = ac \]

4. **Expanding the Equation**:
- Expanding the left side:
\[ \frac{(a + c)^2}{4} = ac \]
- This simplifies to:
\[ (a + c)^2 = 4ac \]

5. **Rearranging the Equation**:
- Expanding the left side:
\[ a^2 + 2ac + c^2 = 4ac \]
- Rearranging gives:
\[ a^2 + c^2 - 2ac = 0 \]

6. **Factoring the Equation**:
- This can be factored as:
\[ (a - c)^2 = 0 \]
- This implies:
\[ a - c = 0 \quad \Rightarrow \quad a = c \]

7. **Conclusion**:
- Since \(a = c\), and substituting back into the A.P. and G.P. conditions, we find that \(b\) must also equal \(a\) (or \(c\)). Thus, we conclude:
\[ a = b = c \]

### Final Statement:
The correct statement is that \(a = b = c\).