Question
If $${a_1},{a_2},{a_3},......,{a_n},......$$ are in G.P., then the value of the determinant \[\left| \begin{array}{l}
\,\,\log {a_n}\,\,\,\,\,\,\,\,\log {a_{n + 1}}\,\,\,\,\,\,\,\,\,\,\log {a_{n + 2}}\\
\log {a_{n + 3}}\,\,\,\,\,\,\,\log {a_{n + 4}}\,\,\,\,\,\,\,\,\,\log {a_{n + 5}}\\
\log {a_{n + 6}}\,\,\,\,\,\,\,\log {a_{n + 7}}\,\,\,\,\,\,\,\,\,\log {a_{n + 8}}
\end{array} \right|,\] is
A.
$$- 2$$
B.
1
C.
2
D.
0
Answer :
0
Solution :
Let $$r$$ be the common ratio, then
\[\left| \begin{array}{l}
\,\,\log {a_n}\,\,\,\,\,\,\,\,\log {a_{n + 1}}\,\,\,\,\,\,\,\,\,\,\log {a_{n + 2}}\\
\log {a_{n + 3}}\,\,\,\,\,\,\,\log {a_{n + 4}}\,\,\,\,\,\,\,\,\,\log {a_{n + 5}}\\
\log {a_{n + 6}}\,\,\,\,\,\,\,\log {a_{n + 7}}\,\,\,\,\,\,\,\,\,\log {a_{n + 8}}
\end{array} \right|\]
\[ = \left| \begin{array}{l}
\log {a_1}{r^{n - 1}}\,\,\,\,\,\,\log {a_1}{r^n}\,\,\,\,\,\,\,\,\,\log {a_1}{r^{n + 1}}\\
\log {a_1}{r^{n + 2}}\,\,\,\,\,\,\log {a_1}{r^{n + 3}}\,\,\,\,\,\,\log {a_1}{r^{n + 4}}\\
\log {a_1}{r^{n + 5}}\,\,\,\,\,\,\log {a_1}{r^{n + 6}}\,\,\,\,\,\,\log {a_1}{r^{n + 7}}
\end{array} \right|\]
\[ = \left| \begin{array}{l}
\log \,{a_1} + \left( {n - 1} \right)\log r\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\log {a_1} + n\log r\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\log {a_1} + \left( {n + 1} \right)\log r\\
\log {a_1} + \left( {n + 2} \right)\log r\,\,\,\,\,\,\,\,\,\,\,\log {a_1} + \left( {n + 3} \right)\log r\,\,\,\,\,\,\,\,\,\,\,\log {a_1} + \left( {n + 4} \right)\log r\\
\log {a_1} + \left( {n + 5} \right)\log r\,\,\,\,\,\,\,\,\,\,\,\log {a_1} + \left( {n + 6} \right)\log r\,\,\,\,\,\,\,\,\,\,\,\log {a_1} + \left( {n + 7} \right)\log r
\end{array} \right|\]
$$ = 0\left[ {{\text{Apply }}{c_2} \to {c_2} - \frac{1}{2}{c_1} - \frac{1}{2}{c_3}} \right]$$