Question

If $a_1,a_2,a_3,...$ are in GP, then the value of the determinant $∆=\left|\begin{array}{ccc}\log a_n& \log a_{n+1}& \log a_{n+2}\\ \log a_{n+3}& \log a_{n+4}& \log a_{n+5}\\ \log a_{n+6}& \log a_{n+7}& \log a_{n+8}\end{array}\right|$

• A. $-2$
• B. $1$
• C. $2$
• D. $0$

Answer 1

The correct option is D

$0$

Finding the value of the determinant $∆=\left|\begin{array}{ccc}\log a_n& \log a_{n+1}& \log a_{n+2}\\ \log a_{n+3}& \log a_{n+4}& \log a_{n+5}\\ \log a_{n+6}& \log a_{n+7}& \log a_{n+8}\end{array}\right|$

Given $a_1,a_2,a_3,...$ are in GP

Therefore $r=\frac{a_2}{a_1}=\frac{a_3}{a_2}=\dots =\frac{a_n}{a_{n-1}}$

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

Apply

$$\begin{gathered} C_3\to C_3-C_2 \\ {C}_2\to C_2-C_1 \end{gathered}$$

Since

$$\begin{gathered} \log a_{n+2}–\log a_{n+1}=\log \left(\frac{a_{n+2}}{a_{n+1}}\right) \\ =\log r \end{gathered}$$

Also $\log a_{n+1}–\log a_n=\log \left(\frac{a_{n+1}}{a_n}\right)$

$=\log r$

So, $∆=\left|\begin{array}{ccc}\log a_n& \log r& \log r\\ \log a_{n+3}& \log r& \log r\\ \log a_{n+6}& \log r& \log r\end{array}\right|$

Two Columns are same

So $∆=0$

Hence, the value of $\mathbf{∆}\mathbf{=}\left|\begin{array}{ccc}log{a}_n& log{a}_{n+1}& log{a}_{n+2}\\ log{a}_{n+3}& log{a}_{n+4}& log{a}_{n+5}\\ log{a}_{n+6}& log{a}_{n+7}& log{a}_{n+8}\end{array}\right|$ is $\mathbf{0}\mathbf{.}$