If `a_(1), a_(2), a_(3), …., a_(9)` are in GP, then what is the value of the following determinant?
`|(ln a_(1), ln a_(2), ln a_(3)),(lna_(4), lna_(5), ln a_(6)), (ln a_(7), ln a_(8), ln a_(9))| `

Options

A. 0
B. 1
C. 2
D. 4

Answer

To solve the problem, we need to find the value of the determinant given that \( a_1, a_2, a_3, \ldots, a_9 \) are in a geometric progression (GP).

1. **Identify the terms in GP**:
Let the first term of the GP be \( a \) and the common ratio be \( r \). Then the terms can be expressed as:
\[ a_1 = a, \quad a_2 = ar, \quad a_3 = ar^2, \quad a_4 = ar^3, \quad a_5 = ar^4, \quad a_6 = ar^5, \quad a_7 = ar^6, \quad a_8 = ar^7, \quad a_9 = ar^8 \]

2. **Take the natural logarithm of each term**:
We will take the natural logarithm of each term:
\[ \ln a_1 = \ln a, \quad \ln a_2 = \ln a + \ln r, \quad \ln a_3 = \ln a + 2\ln r, \quad \ln a_4 = \ln a + 3\ln r \]
\[ \ln a_5 = \ln a + 4\ln r, \quad \ln a_6 = \ln a + 5\ln r, \quad \ln a_7 = \ln a + 6\ln r, \quad \ln a_8 = \ln a + 7\ln r, \quad \ln a_9 = \ln a + 8\ln r \]

3. **Construct the determinant**:
The determinant can then be expressed as:
\[ \begin{vmatrix} \ln a & \ln a + \ln r & \ln a + 2\ln r \\ \ln a + 3\ln r & \ln a + 4\ln r & \ln a + 5\ln r \\ \ln a + 6\ln r & \ln a + 7\ln r & \ln a + 8\ln r \end{vmatrix} \]

4. **Simplify the determinant using properties of logarithms**:
We can rewrite the determinant as:
\[ \begin{vmatrix} \ln a & \ln a + \ln r & \ln a + 2\ln r \\ \ln a + 3\ln r & \ln a + 4\ln r & \ln a + 5\ln r \\ \ln a + 6\ln r & \ln a + 7\ln r & \ln a + 8\ln r \end{vmatrix} = \begin{vmatrix} \ln a & \ln a + \ln r & \ln a + 2\ln r \\ 0 & \ln r & 2\ln r \\ 0 & 2\ln r & 3\ln r \end{vmatrix} \]

5. **Row operations**:
Subtract the first row from the second and third rows:
\[ R_2 \rightarrow R_2 - R_1, \quad R_3 \rightarrow R_3 - R_1 \]
This gives us:
\[ \begin{vmatrix} \ln a & \ln a + \ln r & \ln a + 2\ln r \\ 0 & 3\ln r - \ln a & 2\ln r \\ 0 & 6\ln r - \ln a & 3\ln r \end{vmatrix} \]

6. **Check for proportional rows**:
Notice that the second and third rows are proportional to each other. This means that the determinant will be zero.

7. **Conclusion**:
Therefore, the value of the determinant is:
\[ \boxed{0} \]