Question

If $a_1,a_2,a_3,\cdots ,a_r$ are in $G.P.$ with common ratio $R,$ then the determinant $\left|\begin{array}{ccc}{a}_{r+1}& a_{r+5}& a_{r+9}\\ a_{r+7}& a_{r+11}& a_{r+15}\\ a_{r+11}& a_{r+17}& a_{r+21}\end{array}\right|$ is

• A. Propotional to $R$
• B. Propotional to $R^2$
• C. Independent of $R$
• D. Propotional to $R^3$

Answer 1

The correct option is C Independent of $R$

We know that, $a_{r+1}=A{R}^r$
where, $A=$ first term, $R=$ common ratio
$\Rightarrow \Delta =\left|\begin{array}{ccc}A{R}^r& A{R}^{r+4}& A{R}^{r+8}\\ A{R}^{r+6}& A{R}^{r+10}& A{R}^{r+14}\\ A{R}^{r+10}& A{R}^{r+16}& A{R}^{r+20}\end{array}\right|$
Taking $A{R}^r,A{R}^{r+6}$ and $A{R}^{r+10}$ common from rows $R_1,R_2$ and $R_3$ respectively, we get
$\Rightarrow \Delta =A{R}^r\cdot A{R}^{r+6}\cdot A{R}^{r+10}\left|\begin{array}{ccc}1& R^4& R^8\\ 1& R^4& R^8\\ 1& R^6& R^{10}\end{array}\right|\therefore \Delta =0$
($\because$ rows $R_1$ and $R_2$ are identical.)