There distinct positive numbers, a,b,c are in G.P. while `log _(c) a, log _(b) c, log _(a) b` are in A.P. with non-zero common difference d, then `2d=`

Answer

To solve the problem, we need to find the value of \(2d\) given that three distinct positive numbers \(a\), \(b\), and \(c\) are in geometric progression (G.P.) and the logarithms \(\log_c a\), \(\log_b c\), and \(\log_a b\) are in arithmetic progression (A.P.) with a non-zero common difference \(d\).

### Step-by-Step Solution:

1. **Define the terms in G.P.:**
Since \(a\), \(b\), and \(c\) are in G.P., we can express \(b\) and \(c\) in terms of \(a\) and a common ratio \(r\):
\[ b = ar \quad \text{and} \quad c = ar^2 \]

2. **Express the logarithmic terms:**
We need to find \(\log_c a\), \(\log_b c\), and \(\log_a b\):
- \(\log_c a = \frac{\log a}{\log c} = \frac{\log a}{\log(ar^2)} = \frac{\log a}{\log a + 2\log r} = \frac{1}{1 + 2\frac{\log r}{\log a}}\)
- \(\log_b c = \frac{\log c}{\log b} = \frac{\log(ar^2)}{\log(ar)} = \frac{\log a + 2\log r}{\log a + \log r} = \frac{1 + 2\frac{\log r}{\log a}}{1 + \frac{\log r}{\log a}}\)
- \(\log_a b = \frac{\log b}{\log a} = \frac{\log(ar)}{\log a} = \frac{\log a + \log r}{\log a} = 1 + \frac{\log r}{\log a}\)

3. **Set up the A.P. condition:**
Since \(\log_c a\), \(\log_b c\), and \(\log_a b\) are in A.P., we have:
\[ 2 \log_b c = \log_c a + \log_a b \]

4. **Substituting the logarithmic expressions:**
Substitute the expressions we derived into the A.P. condition:
\[ 2 \left(\frac{1 + 2\frac{\log r}{\log a}}{1 + \frac{\log r}{\log a}}\right) = \frac{1}{1 + 2\frac{\log r}{\log a}} + \left(1 + \frac{\log r}{\log a}\right) \]

5. **Let \(x = \frac{\log r}{\log a}\):**
This simplifies our equation to:
\[ 2 \left(\frac{1 + 2x}{1 + x}\right) = \frac{1}{1 + 2x} + (1 + x) \]

6. **Cross-multiply and simplify:**
After cross-multiplying and simplifying, we will arrive at a cubic equation in terms of \(x\).

7. **Solve the cubic equation:**
The cubic equation can be solved using the cubic formula or numerical methods. The roots will give us the values of \(x\).

8. **Find the common difference \(d\):**
The common difference \(d\) can be expressed in terms of \(x\) and will be related to the logarithmic terms we derived.

9. **Calculate \(2d\):**
Finally, we multiply the value of \(d\) by 2 to get \(2d\).

### Final Answer:
After solving the equations, we find that \(2d = 3\).