Question

If $a_1,a_2,a_3,...$ from a geometric progression and $a_i>0$ for all $i\ge 1$, then $\left|\begin{array}{ccc}\log a_m& \log a_{m+1}& \log a_{m+2}\\ \log a_{m+3}& \log a_{m+4}& \log a_{m+5}\\ \log a_{m+6}& \log a_{m+7}& \log a_{m+8}\end{array}\right|$ is equal to

• A. $\log a_{m+8}-\log a_m$
• B. $\log a_m$
• C. $2\log a_{m+1}$
• D. $0$

Answer 1

Answer: (D)

$0$

Since $a_1,a_2,a_3....$ are in G.P
$\therefore \log a_1,\log a_2,\log a_3,....$ are in G.P
$\Rightarrow \log a_m+\log a_{m+2}=2\log a_{m+1}$
$\log a_{m+3}+\log a_{m+5}=2\log a_{m+4}$
$\log a_{m+6}+\log a_{m+8}=2\log a_{m+7}$
applying $C_2\to C_2-\left(1/2\right)(C_1+C_3)$ we get
$\Delta =\left|\begin{array}{ccc}\log a_m& 0& \log a_{m+2}\\ \log a_{m+3}& 0& \log a_{m+5}\\ \log a_{m+6}& 0& \log a_{m+8}\end{array}\right|=0$