Question

If $a_1,a_2,a_3,...a_{n,}...$ are in G.P., 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. $1$
• B. $0$
• C. $4$
• D. $2$

Answer 1

The correct option is B $0$

$a_1,a_2,a_3{a}_n\to$G.P

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

$a_n=a_1{r}^{n-1}$

$a_{n+1}=a_1{r}^n$

$a_{n+2}=a_1{r}^{n+1}$

$a_{n+3}=a_1{r}^{n+2}$

$\Rightarrow △=\left|\begin{array}{ccc}{a}_1{r}^{n-1}& a_1{r}^n& a_1{r}^{n+1}\\ a_1{r}^{n+2}& a_1{r}^{n+3}& a_1{r}^{n+4}\\ a_1{r}^{n+5}& a_1{r}^{n+6}& a_1{r}^{n+7}\end{array}\right|$

$=\left(a_1{r}^{n-1}\right)\left(a_1{r}^n\right)\left(a_1{r}^{n+1}\right)\left|\begin{array}{ccc}1& 1& 1\\ r^3& r^3& r^3\\ r^6& r^6& r^6\end{array}\right|$

$\Rightarrow △=0$