Question

If $a_1,a_2,a_3,...,a_n,...$are in $GP$ then the determinant $\Delta =\left|\begin{array}{ccc}log{a}_n& log{a}_{n+1}& log{a}_{n+2}\\ log{a}_{n+3}& log{a}_{n+4}& log{a}_{n+5}\\ log{a}_{n+6}& log{a}_{n+7}& log{a}_{n+8}\end{array}\right|$ is equal to

• A. $0$
• B. $1$
• C. $2$
• D. $4$

Answer 1

The correct option is A $0$

$\Delta =\left|\begin{array}{ccc}log{a}_n& log{a}_{n+1}& log{a}_{n+2}\\ log{a}_{n+3}& log{a}_{n+4}& log{a}_{n+5}\\ log{a}_{n+6}& log{a}_{n+7}& log{a}_{n+8}\end{array}\right|$
$=\left|\begin{array}{ccc}loga{r}^{n-1}& loga{r}^n& loga{r}^{n+1}\\ loga{r}^{n+2}& loga{r}^{n+3}& loga{r}^{n+4}\\ loga{r}^{n+5}& loga{r}^{n+6}& loga{r}^{n+7}\end{array}\right|$
$\Delta =\left|\begin{array}{ccc}loga+(n-1)\log r& loga+n\log r& loga+(n+1)\log r\\ loga+(n+2)\log r& loga+(n+3)\log r& loga+(n+4)\log r\\ loga+(n+4)\log r& loga+(n+6)\log r& loga+(n+7)\log r\end{array}\right|$
$C_1\to C_1-C_2,C_2\to C_2-C_3$
$\Delta =\left|\begin{array}{ccc}-\log r& -\log r& loga+(n+1)\log r\\ -\log r& -\log r& loga+(n+4)\log r\\ -\log r& -\log r& loga+(n+7)\log r\end{array}\right|$
$\Rightarrow \Delta =0$