If log a , log b , log c are in A.P. then

Options

A. `a,b,c` are in G.P.
B. `a^2,b^2,c^2` are in G.P.
C. `a,b,c` are in A.P.
D. none of these

Answer

To solve the problem, we need to analyze the condition that \( \log A, \log B, \log C \) are in Arithmetic Progression (A.P.).

### Step-by-Step Solution:

1. **Understanding A.P.**:
If \( \log A, \log B, \log C \) are in A.P., then by definition, we have:
\[ 2 \log B = \log A + \log C \]

2. **Using Logarithmic Properties**:
We can use the property of logarithms that states:
\[ \log A + \log C = \log(AC) \]
Thus, we can rewrite the equation as:
\[ 2 \log B = \log(AC) \]

3. **Exponentiating Both Sides**:
To eliminate the logarithm, we exponentiate both sides:
\[ \log B^2 = \log(AC) \]
This implies:
\[ B^2 = AC \]

4. **Interpreting the Result**:
The equation \( B^2 = AC \) indicates that the square of the middle term \( B \) is equal to the product of the first term \( A \) and the last term \( C \). This is the condition for the numbers \( A, B, C \) to be in Geometric Progression (G.P.).

5. **Conclusion**:
Therefore, we conclude that \( A, B, C \) are in G.P.

6. **Checking the Options**:
- **Option 1**: \( A, B, C \) are in A.P. (Incorrect)
- **Option 2**: \( A^2, B^2, C^2 \) are in A.P. (Correct, since squaring maintains the G.P. property)
- **Option 3**: \( A, B, C \) are in G.P. (Correct)
- **Option 4**: None of these (Incorrect)

Thus, the correct options are that \( A, B, C \) are in G.P. and \( A^2, B^2, C^2 \) are also in G.P.