Let $a_{1}, a_{2}, a_{3}, \ldots, a_{10}$ be in G.P. with $a_{i} > 0$ for $i = 1, 2, \ldots, 10$ and $S$ be…
Let $a_{1}, a_{2}, a_{3}, \ldots, a_{10}$ be in G.P. with $a_{i} > 0$ for $i = 1, 2, \ldots, 10$ and $S$ be the set of pairs $(r, k)$, $r, k \in \mathbb{N}$ (the set of natural numbers) for which
$\begin{aligned}
\left|
\begin{array}{ccc}
\log_{e} a_{1}^{r} a_{2}^{k} & \log_{e} a_{2}^{r} a_{3}^{k} & \log_{e} a_{3}^{r} a_{4}^{k} \\
\log_{e} a_{4}^{r} a_{5}^{k} & \log_{e} a_{5}^{r} a_{6}^{k} & \log_{e} a_{6}^{r} a_{7}^{k} \\
\log_{e} a_{7}^{r} a_{8}^{k} & \log_{e} a_{8}^{r} a_{9}^{k} & \log_{e} a_{9}^{r} a_{10}^{k}
\end{array}
\right|
= 0
\end{aligned}$
Then the number of elements in $S$ is:
Infinitely many
Solution
$C_3 \rightarrow C_3 - C_2$, $C_2 \rightarrow C_2 - C_1$ (Let $\alpha$ is common ratio of $GP$)
$\begin{aligned}
\begin{vmatrix}
\log_e a_1^r a_2^k & \log_e |\alpha^{r+k}| & \log_e |\alpha^{r+k}| \\
\log_e a_4^r a_5^k & \log_e |\alpha^{r+k}| & \log_e |\alpha^{r+k}| \\
\log_e a_7^r a_8^k & \log_e |\alpha^{r+k}| & \log_e |\alpha^{r+k}|
\end{vmatrix}
= 0
\end{aligned}$
which is always true because $C_2$ and $C_3$ are identical.