If a_1 , a_2 , a_3 ..... a_n .... are in G.P. then the value of t… - Vidyadip

MCQ

If ${a_1},{a_2},{a_3}.....{a_n}....$ are in $G.P.$ then the value of the determinant $\left| {\,\begin{array}{*{20}{c}}{\log {a_n}}&{\log {a_{n + 1}}}&{\log {a_{n + 2}}}\\{\log {a_{n + 3}}}&{\log {a_{n + 4}}}&{\log {a_{n + 5}}}\\{\log {a_{n + 6}}}&{\log {a_{n + 7}}}&{\log {a_{n + 8}}}\end{array}\,} \right|$ is

A
$-2$
B
$1$
C
$2$
✓
$0$

✓

Answer

Correct option: D.

$0$

d
(d) We have ${a_1},\,{a_2},\,{a_3}$..... an in $G.P.$

then $r = \frac{{{a_2}}}{{{a_1}}}$ i.e., $r = \frac{{{a_{n + 1}}}}{{{a_n}}} = \frac{{{a_{n + 2}}}}{{{a_{n + 1}}}} = .....$..

Hence ${\log _r} = \log ({a_{n + 1}}) - \log ({a_n}) = \log ({a_{n + 2}}) - \log ({a_{n + 1}}) = ...$

Now $\left| {\,\begin{array}{*{20}{c}}{\log {a_n}}&{\log {a_{n + 1}}}&{\log {a_{n + 2}}}\\{\log {a_{n + 3}}}&{\log {a_{n + 4}}}&{\log {a_{n + 5}}}\\{\log {a_{n + 6}}}&{\log {a_{n + 7}}}&{\log {a_{n + 8}}}\end{array}\,} \right|$

Operate ${C_2} \to {C_2} - {C_1}$ and ${C_3} \to {C_3} - {C_2}$

= $\left| {\,\begin{array}{*{20}{c}}{\log {a_n}}&{(\log {a_{n + 1}} - \log {a_n})}&{(\log {a_{n + 2}} - \log {a_{n + 1}})}\\{\log {a_{n + 3}}}&{(\log {a_{n + 4}} - \log {a_{n + 3}})}&{(\log {a_{n + 5}} - \log {a_{n + 4}})}\\{\log {a_{n + 6}}}&{(\log {a_{n + 7}} - \log {a_{n + 6}})}&{(\log {a_{n + 8}} - \log {a_{n + 7}})}\end{array}\,} \right|$

= $\left| {\,\begin{array}{*{20}{c}}{\log {a_n}}&{\log r}&{\log r}\\{\log {a_{n + 3}}}&{\log r}&{\log r}\\{\log {a_{n + 6}}}&{\log r}&{\log r}\end{array}\,} \right|{\rm{ }} = {\rm{ 0}}$.

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