Question

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

Answer 1

Let $r$ be the common ratio of the given G.P. Now, $\left|\begin{array}{ccc}\log a_n& \log a_{n+1}& \log a_{n+2}\\ \log a_{n+3}& \log a_{n+4}& \log a_{n+5}\\ \log a_{n+6}& \log a_{n+7}& \log a_{n+8}\end{array}\right|$

$[C_2\to C_2-C_1$ and $C_3\to C_3-C_2]$

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

$=\left|\begin{array}{ccc}\log a_n& \log r& \log r\\ \log a_{n+3}& \log r& \log r\\ \log a_{n+6}& \log r& \log r\end{array}\right|$
Since $\frac{a_{p+1}}{a_p}=r$ and $\log a_{p+1}-\log a_p=\log \frac{a_{p+1}}{a_p}=\log r,$ hence, determinant value $=0[$ Since $C_3=C_2]$