Question
If $${a_1},{a_2},{a_3},.....,{a_n},.....$$ are G.P., then the determinant \[\Delta = \left| \begin{array}{l}
\,\,\log {a_n}\,\,\,\,\,\,\,\,\,\,\log {a_{n + 1}}\,\,\,\,\,\,\,\,\,\log {a_{n + 2}}\\
\log {a_{n + 3}}\,\,\,\,\,\,\,\,\,\log {a_{n + 4}}\,\,\,\,\,\,\,\,\log {a_{n + 5}}\\
\log {a_{n + 6}}\,\,\,\,\,\,\,\,\,\log {a_{n + 7}}\,\,\,\,\,\,\,\,\log {a_{n + 8}}
\end{array} \right|\] is equal to
A.
1
B.
0
C.
4
D.
2
Answer :
0
Solution :
$$\because \,\,{a_1},{a_2},{a_3},.....,$$ are in G.P.
∴ Using $${a_n} = a{r^{n - 1}},$$ we get the given determinant, as
\[\left| \begin{array}{l}
\log a{r^{n - 1}}\,\,\,\,\,\,\,\,\,\log a{r^n}\,\,\,\,\,\,\,\,\,\,\,\,\log a{r^{n + 1}}\\
\log a{r^{n + 2}}\,\,\,\,\,\,\,\,\,\log a{r^{n + 3}}\,\,\,\,\,\,\,\,\log a{r^{n + 4}}\\
\log a{r^{n + 5}}\,\,\,\,\,\,\,\,\,\log a{r^{n + 6}}\,\,\,\,\,\,\,\,\log a{r^{n + 7}}
\end{array} \right|\]
$${\text{Operating }}{C_3} - {C_2}\,{\text{and }}{C_2} - {C_1}$$ and using
$$\log m - \log n = \log \frac{m}{n}\,\,{\text{we get}}$$
\[ = \left| \begin{array}{l}
\log a{r^{n - 1}}\,\,\,\,\,\,\,\,\,\log r\,\,\,\,\,\,\,\,\log r\\
\log a{r^{n + 2}}\,\,\,\,\,\,\,\,\,\log r\,\,\,\,\,\,\,\,\log r\\
\log a{r^{n + 5}}\,\,\,\,\,\,\,\,\,\log r\,\,\,\,\,\,\,\,\log r
\end{array} \right|\]
= 0 (two columns being identical)