Question

If $a_1, a_2, a_3,......, a_n $,.... are in G.P., then the value of the determinant $\begin{vmatrix}\log a_{n}& \log a_{n+1}&\log a_{n+2}\\ \log a_{n+3}& \log a_{n+4}&\log a_{n+5}\\ \log a_{n+6} &\log a_{n+7}& \log a_{n+8}\end{vmatrix} $, is

• A. -2
• B. 1
• C. 2
• D. 0

Answer 1

Answer: (D)

Let r be the common ratio, then
$\begin{vmatrix}\log a_{n}& \log a_{n+1}&\log a_{n+2}\\ \log a_{n+3}& \log a_{n+4}&\log a_{n+5}\\ \log a_{n+6} &\log a_{n+7}& \log a_{n+8}\end{vmatrix} $

$= \begin{vmatrix}\log a_{1}r^{n-1}&\log a_{1}r^{n}&\log a_{1}r^{n+1}\\ \log a_{1}r^{n+2} &\log a_{1}r^{n+3}&\log a_{1}r^{n+4}\\ \log a_{1}r^{n+5} &\log a_{1}r^{n+6}&\log a_{1}r^{n+7}\end{vmatrix}$

$ = \begin{vmatrix}\log a_{1} +\left(n-1\right)\log r&\log a_{1}+n \log r&\log a_{1}\left(n+1\right)\log r\\ \log a_{1}+\left(n+2\right)\log r & \log a_{1} +\left(n+3\right)\log r&\log a_{1}+\left(n+4\right)\log r\\ \log a_{1}+\left(n+5\right)\log r &\log a_{1}+\left(n+6\right)\log r&\log a_{2} +\left(n+7\right)\log r \end{vmatrix}$

$= 0 \left[ \text{Apply}\, c_2 \to c_2 - \frac{1}{2} c_1 - \frac{1}{2} c_3 \right]$