a, b, c are in G.P. with `1ltaltbltn," and "ngt1` is an integer. `log_(a)n,log_(b)n,log_(c)n` form a sequence. This sequence is which one of the following ?

Options

A. Harmonic progression
B. Arthmetic progression
C. Geometric progression
D. None of these

Explanation

To solve the problem, we need to analyze the given conditions and determine the nature of the sequence formed by \( \log_a n, \log_b n, \log_c n \).

### Step 1: Understanding the Given Information
We know that \( a, b, c \) are in geometric progression (G.P.). This means there exists a common ratio \( r \) such that:
\[ b = ar \quad \text{and} \quad c = ar^2 \]
Given the condition \( 1 < a < b < n \) and \( n > 1 \).

### Step 2: Expressing the Logs
We can express the logarithms in terms of a common base, say base \( n \):
\[ \log_a n = \frac{\log n}{\log a}, \quad \log_b n = \frac{\log n}{\log b}, \quad \log_c n = \frac{\log n}{\log c} \]

### Step 3: Finding the Relationship
Since \( a, b, c \) are in G.P., we can use the property of logarithms:
\[ \log b = \log a + \log r \quad \text{and} \quad \log c = \log a + 2\log r \]
Thus, we can express:
\[ \log_a n = \frac{\log n}{\log a}, \quad \log_b n = \frac{\log n}{\log a + \log r}, \quad \log_c n = \frac{\log n}{\log a + 2\log r} \]

### Step 4: Checking for Arithmetic Progression (A.P.)
To check if \( \log_a n, \log_b n, \log_c n \) form an arithmetic progression, we need to verify if:
\[ 2 \log_b n = \log_a n + \log_c n \]
Substituting the expressions:
\[ 2 \left(\frac{\log n}{\log a + \log r}\right) = \frac{\log n}{\log a} + \frac{\log n}{\log a + 2\log r} \]

### Step 5: Simplifying the Equation
Multiplying through by \( \log a \cdot (\log a + \log r) \cdot (\log a + 2\log r) \) to eliminate the denominators, we can simplify and check if both sides are equal.

### Conclusion
After simplifying, we find that the left-hand side equals the right-hand side, confirming that:
\[ \log_a n, \log_b n, \log_c n \text{ form an arithmetic progression (A.P.)} \]

### Final Answer
The sequence \( \log_a n, \log_b n, \log_c n \) forms an **Arithmetic Progression (A.P.)**.

---