For a natural number $n$, $f ( n )$ is defined as follows: $$f ( n ) = \begin{cases} \log _ { 3 } n & ( n \text{ is odd} ) \\ \log _ { 2 } n & ( n \text{ is even} ) \end{cases}$$ For the sequence $\left\{ a _ { n } \right\}$ where $a _ { n } = f \left( 6 ^ { n } \right) - f \left( 3 ^ { n } \right)$, what is the value of $\sum _ { n = 1 } ^ { 15 } a _ { n }$? [3 points] (1) $120 \left( \log _ { 2 } 3 - 1 \right)$ (2) $105 \log _ { 3 } 2$ (3) $105 \log _ { 2 } 3$ (4) $120 \log _ { 2 } 3$ (5) $120 \left( \log _ { 3 } 2 + 1 \right)$
For a natural number $n$, $f ( n )$ is defined as follows:
$$f ( n ) = \begin{cases} \log _ { 3 } n & ( n \text{ is odd} ) \\ \log _ { 2 } n & ( n \text{ is even} ) \end{cases}$$
For the sequence $\left\{ a _ { n } \right\}$ where $a _ { n } = f \left( 6 ^ { n } \right) - f \left( 3 ^ { n } \right)$, what is the value of $\sum _ { n = 1 } ^ { 15 } a _ { n }$? [3 points]\\
(1) $120 \left( \log _ { 2 } 3 - 1 \right)$\\
(2) $105 \log _ { 3 } 2$\\
(3) $105 \log _ { 2 } 3$\\
(4) $120 \log _ { 2 } 3$\\
(5) $120 \left( \log _ { 3 } 2 + 1 \right)$