Question

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

• A. $0$
• B. $1$
• C. $2$
• D. $4$

Answer 1

Answer: (A)

$0$

Let $r$ be the common ration of the given G.P.

Now,

$\left|\begin{array}{ccc}\log a_n& \log a_{n+1}& \log a_{n+2}\\ \log a_{n+3}& \log a_{n+4}& \log a_{n+5}\\ \log a_{n+6}& \log a_{n+7}& \log a_{n+8}\end{array}\right|$

[$C_2^{\prime }=C_2-C_1$ and $C_3^{\prime }=C_3-C_2$]

$=\left|\begin{array}{ccc}\log a_n& \log a_{n+1}-\log a_n& \log a_{n+2}-\log a_{n+1}\\ \log a_{n+3}& \log a_{n+4}-\log a_{n+3}& \log a_{n+5}-\log a_{n+4}\\ \log a_{n+6}& \log a_{n+7}-\log a_{n+6}& \log a_{n+8}-\log a_{n+7}\end{array}\right|$

$=\left|\begin{array}{ccc}\log a_n& \log r& \log r\\ \log a_{n+3}& \log r& \log r\\ \log a_{n+6}& \log r& \log r\end{array}\right|$ [Since $\frac{a_{p+1}}{a_p}=r$ and $\log a_{p+1}-\log a_p$$=\log \frac{a_{p+1}}{a_p}=\log r$]

$=0$ [Since $C_3=C_2$]