Question

If $a_1,a_2,a_3...a_n\in GP$ then the value of the determinant $△=\left|\begin{array}{ccc}\log a_n& \log a_{n+1}& \log a_{n+2}\\ \log a_{n+3}& \log a_{n+4}& \log a_{n+5}\\ \log a_{n+6}& \log a_{n+7}& \log a_{n+8}\end{array}\right|$ equals

• A. 2
• B. 1
• C. 0
• D. -2

Answer 1

The correct option is C 0

$△=\left|\begin{array}{ccc}\log a_n& \log a_{n+1}& \log a_{n+2}\\ \log a_{n+3}& \log a_{n+4}& \log a_{n+3}\\ \log a_{n+6}& \log a_{n+7}& \log a_{n+4}\end{array}\right|$

$a_1,a_2{a}_n$are in GP

$\log a_1,\log a_2...$are in AP

$\Rightarrow △=\left|\begin{array}{ccc}a& a+d& a+2d\\ a+3d& a+4d& a+5d\\ a+6d& a+7d& a+8d\end{array}\right|$

$C_2\to C_2-C_1,C_3\to C_3-C_1$

$△=\left|\begin{array}{ccc}a& d& 2d\\ a+3d& d& 2d\\ a+6d& d& 2d\end{array}\right|$

$C_2\&C_3$ are similar

$\Rightarrow △=0$