Let the sequence $\left\{ a _ { n } \right\} ( n = 1,2,3 , \cdots )$ be an arithmetic progression satisfying
$$a _ { 2 } = 2 , \quad a _ { 6 } = 3 a _ { 3 } .$$
Then, consider the series $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } } { r ^ { a _ { n } } }$, where $r$ is a positive real number.
(1) When we denote the first term of $\left\{ a _ { n } \right\}$ by $a$, and the common difference by $d$, we have
$$a = \mathbf { A B } , \quad d = \mathbf { C } .$$
(2) The series $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } } { r ^ { a _ { n } } }$ is an infinite geometric series where the first term is $\square r^{\mathbf{E}}$, and the common ratio is $\dfrac { \mathbf { F } } { r^{\mathbf{G}} }$. Hence, this series converges when
$$r > 3 ^ { \frac { \mathbf { H } } { \mathbf{I} } } ,$$
and its sum $S$ is
$$S = \frac { \square \mathbf { J } \, r ^ { \mathbf { K } } } { r ^ { \mathbf { L } } - \mathbf { M } } .$$
(3) This sum $S$ is minimized at
$$r = \mathbf { N } ^ { \frac { \mathbf { O } } { 2 } } .$$