If lna, lnb, lnc are in A.P and lna-ln2b, ln2b-ln2b-ln3c, ln3c-lna are in A.P, then a:b:c is

Options

A. `1:2:3`
B. `7:7:4`
C. `9:6:4`
D. `4:6:9`

Answer

To solve the problem, we need to analyze the given conditions step by step.

### Step 1: Understand the first condition
We are given that \( \ln a, \ln b, \ln c \) are in Arithmetic Progression (A.P). This means that:
\[ 2 \ln b = \ln a + \ln c \]
Using properties of logarithms, we can rewrite this as:
\[ \ln b^2 = \ln(ac) \implies b^2 = ac \tag{1} \]

### Step 2: Understand the second condition
Next, we are given that \( \ln a - \ln 2b, \ln 2b - \ln 3c, \ln 3c - \ln a \) are also in A.P. This means:
\[ 2(\ln 2b) = (\ln a + \ln 3c) \]
Using properties of logarithms, we can rewrite this as:
\[ 2(\ln 2 + \ln b) = \ln a + \ln 3 + \ln c \implies 2\ln 2 + 2\ln b = \ln a + \ln 3 + \ln c \]
Rearranging gives:
\[ 2\ln b - \ln a - \ln c = \ln 3 - 2\ln 2 \tag{2} \]

### Step 3: Substitute from (1) into (2)
From equation (1), we have \( ac = b^2 \). We can substitute \( c = \frac{b^2}{a} \) into equation (2):
\[ 2\ln b - \ln a - \ln\left(\frac{b^2}{a}\right) = \ln 3 - 2\ln 2 \]
This simplifies to:
\[ 2\ln b - \ln a - \left(2\ln b - \ln a\right) = \ln 3 - 2\ln 2 \]
Thus, we have:
\[ 0 = \ln 3 - 2\ln 2 \implies \ln 3 = 2\ln 2 \implies 3 = 2^2 \implies 3 = 4 \]
This is a contradiction, indicating we need to go back and analyze our steps.

### Step 4: Find the ratios
From the first condition \( b^2 = ac \), we can express \( a, b, c \) in terms of a common variable. Let:
\[ a = k, \quad b = m, \quad c = \frac{m^2}{k} \]
Now we need to find the ratio \( a:b:c \).

Using the relationship \( 2b = 3c \) (derived from the second condition), we can substitute \( c \):
\[ 2m = 3\left(\frac{m^2}{k}\right) \implies 2mk = 3m^2 \implies 2k = 3m \implies k = \frac{3}{2}m \]
Now substituting \( k \) back gives:
\[ a = \frac{3}{2}m, \quad b = m, \quad c = \frac{m^2}{\frac{3}{2}m} = \frac{2m}{3} \]

### Step 5: Final ratio
Now we can express the ratio \( a:b:c \):
\[ a:b:c = \frac{3}{2}m : m : \frac{2}{3}m \]
Dividing through by \( m \):
\[ a:b:c = \frac{3}{2} : 1 : \frac{2}{3} \]
To eliminate the fractions, multiply by 6:
\[ a:b:c = 9 : 6 : 4 \]

### Conclusion
Thus, the final ratio \( a:b:c \) is:
\[ \boxed{9:6:4} \]