Question

If $a_1,a_2,a_3,......,a_n$,.... are in G.P., 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. -2
• B. 1
• C. 2
• D. 0

Answer 1

Answer: (D)

Let r be the common ratio, then
$\left|\begin{array}{ccc}\log a_n& \log a_{n+1}& \log a_{n+2}\\ \log a_{n+3}& \log a_{n+4}& \log a_{n+5}\\ \log a_{n+6}& \log a_{n+7}& \log a_{n+8}\end{array}\right|$

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

$=\left|\begin{array}{ccc}\log a_1+\left(n-1\right)\log r& \log a_1+n\log r& \log a_1\left(n+1\right)\log r\\ \log a_1+\left(n+2\right)\log r& \log a_1+\left(n+3\right)\log r& \log a_1+\left(n+4\right)\log r\\ \log a_1+\left(n+5\right)\log r& \log a_1+\left(n+6\right)\log r& \log a_2+\left(n+7\right)\log r\end{array}\right|$

$=0\left[\text{Apply}{c}_2\to c_2-\frac{1}{2}{c}_1-\frac{1}{2}{c}_3\right]$