Question

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

• A. 0
• B. 1
• C. 2
• D. $-2$

Answer 1

Answer: (A)

Since $a_1,a_2,......,a_n$ are in GP Then, $a_n=a_1{r}^{n-1}$ $\Rightarrow$ $\log a_n=\log a_1+(n-1)\log r$ $a_{n+1}=a_1{r}^n$ $\Rightarrow$ $\log a_{n+1}=\log a_1+n\log r$ $a_{n+2}=a_1{r}^n$ $\Rightarrow$ $\log a_{n+2}=\log a_1+(n+1)\log r$ ????.. ????. ????. $a_{n+8}=a_1{r}^{n+7}$ $\Rightarrow$ $\log a_{n+8}=\log a_1+(n+7)\log r$ Now, $\left|\begin{array}{ccc}\log a_n& \log a_{n+1}& \log a_{n+2}\\ \log a_{n+3}& \log a_{n+4}& \log a_{n+5}\\ \log a_{n+6}& \log a_{n+7}& \log a_{n+8}\end{array}\right|$ $=|\begin{array}{cc}\log a_1+(n-1)\log r& \log a_1+n\log r\\ \log a_1+(n+2)\log r& \log a_1+(n+3)\log r\\ \log a_1+(n+5)\log r& \log a_1+(a+6)\log r\end{array}$ $\begin{array}{c}\log a_1+(n+1)\log r\\ \log a_1+(n+4)\log r\\ \log a_1+(n+7)\log r\end{array}|$ Now, $R_2\to R_2-R_1$ and $R_3\to R_3-R_1$ $\Rightarrow$ $|\begin{array}{cc}\log a_1+(n-1)\log r& \log a_1+n\log r\\ 3\log r& 3\log r\\ 3\log r& 3\log r\end{array}$ $\begin{array}{c}\log a_1+(n+1)\log r\\ 3\log r\\ 3\log r\end{array}|=0$ (since two rows are identical)