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. $0$
• B. $1$
• C. $2$
• D. $4$

Answer 1

Answer: (A)

$0$

Let $a_r$ denote the $r^{th}$term of a GP. with first term $a$ and common ratio R
$\therefore a_r=a{R}^{r-1}$
$\therefore \log a_r=\log a+(r-1)\log R$
Now, $\Delta =\left|\begin{array}{ccc}\log a_n& \log a_{n+1}& \log a_{n+2}\\ \log a_{n+3}& \log a_{n+4}& \log a_{n+5}\\ \log a_{n+6}& \log a_{n+7}& \log a_{n+8}\end{array}\right|$

$=\left|\begin{array}{ccc}\log a+(n-1)\log R& \log a+n\log R& \log a+(n+1)\log R\\ \log a+(n+2)\log R& \log a+(n+3)\log R& \log a+(n+4)\log R\\ \log a+(n+5)\log R)& \log a+(n+6)\log R& \log a+(n+7)\log R\end{array}\right|$
Applying $C_2\to 2C_2$ and $C_2\to C_2-(C_1+C_3)$, we get
$\Delta =\frac{1}{2}\left|\begin{array}{ccc}\log a+(r-1)\log R& 0& \log a+(r+1)\log R\\ \log a+(r+2)\log R& 0& \log a+(r+4)\log R\\ \log a+(r+5)\log R)& 0& \log a+(r+7)\log R\end{array}\right|$
$=\frac{1}{2}\times 0=0$