Question

If $a_1,a_2,a_3,...$ are in GP then the values of the determinant $\left|\begin{array}{ccc}log{a}_1& \log a_{n+1}& \log a_{2n+1}\\ log{a}_2& \log a_{n+2}& \log a_{2n+2}\\ log{a}_3& \log a_{n+3}& \log a_{2n+3}\end{array}\right|$ is

• A. $0$
• B. $1$
• C. $\log a_1$
• D. $\log a_2-\log a_1$

Answer 1

Answer: (A)

$0$

$a_1,a_2$-----GP

$\left|\begin{array}{ccc}\log a_1& \log a_{n+1}& \log a_{2n+1}\\ \log a_2& \log a_{n+2}& \log a_{2n+2}\\ \log a_3& \log a_{n+3}& \log a_{2n+3}\end{array}\right|$

$\Rightarrow \left|\begin{array}{ccc}\log a_1& \log a{r}^n& \log a{r}^n\\ \log a_{1}r& \log a{r}^{n+1}& \log a{r}^{2n+1}\\ \log a_1{r}^2& \log a{r}^{n+2}& \log a{r}^{2n+2}\end{array}\right|$

The three rows are in ratio i.e., $R_1=k{R}_2$ or$R_1=l{R}_3$

$det=0$