If in an AP, `t_1 = log_10 a, t_(n+1) = log_10 b and t_(2n+1) = log_10 c` then `a, b, c` are in

Options

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

Answer

To solve the problem, we need to determine the relationship between \( a \), \( b \), and \( c \) given that \( t_1 = \log_{10} a \), \( t_{n+1} = \log_{10} b \), and \( t_{2n+1} = \log_{10} c \) are in an arithmetic progression (AP).

### Step-by-Step Solution:

1. **Understanding the Terms in AP**:
In an arithmetic progression, the difference between consecutive terms is constant. Therefore, we can express the terms as:
\[ t_{n+1} - t_1 = t_{2n+1} - t_{n+1} \]

2. **Substituting the Values**:
Substitute the given values into the AP condition:
\[ \log_{10} b - \log_{10} a = \log_{10} c - \log_{10} b \]

3. **Rearranging the Equation**:
Rearranging gives:
\[ \log_{10} b - \log_{10} a + \log_{10} b = \log_{10} c \]
This simplifies to:
\[ 2\log_{10} b = \log_{10} c + \log_{10} a \]

4. **Using Logarithmic Properties**:
We can use the property of logarithms that states \( \log_{10} x + \log_{10} y = \log_{10}(xy) \):
\[ 2\log_{10} b = \log_{10}(ac) \]

5. **Exponentiating Both Sides**:
To eliminate the logarithm, we exponentiate both sides:
\[ b^2 = ac \]

6. **Conclusion**:
The equation \( b^2 = ac \) indicates that \( a, b, c \) are in a geometric progression (GP).

### Final Answer:
Thus, \( a, b, c \) are in **Geometric Progression (GP)**.

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