question_answer

If\[{{a}_{1}},{{a}_{2}},{{a}_{3}}....,{{a}_{n}},...\]are in GP, then the value of the determinant \[\left| \begin{matrix}    \log {{a}_{n}} & \log {{a}_{n+1}} & \log {{a}_{n+2}}  \\    \log \,\,{{a}_{n+3}} & \log {{a}_{n+4}} & \log {{a}_{n+5}}  \\    \log {{a}_{n+6}} & \log {{a}_{n+7}} & \log {{a}_{n+8}}  \\ \end{matrix} \right|,\] is

A) 0       

B)                                        1                             

C) 2                             

D) \[-2\]

Correct Answer: A

Solution :

Sinceare in GP. Then,    \[\log \,{{a}_{n}}\,=\log \,{{a}_{1}}\,+(n-1)\,\log \,r\]                                                     ??????                 ??????                 ??????                           Now,                                                 Applying,and                                                   (since, two rows are identical) Alternate Solution Since,are in GP, then are in Ap. Given that where, a and c/ are the first term and common difference of an AP. Applying  (since, two columns are identical)