Q1 We are to find the general term $a _ { n }$ of the sequence $\left\{ a _ { n } \right\}$ which is determined by the recurrence formula
$$a _ { 1 } = 18 , \quad a _ { n + 1 } - 12 a _ { n } + 3 ^ { n + 2 } = 0 \quad ( n = 1,2,3 , \cdots ) .$$
When we define a sequence $\left\{ b _ { n } \right\}$ by
$$b _ { n } = \frac { a _ { n } } { \mathbf{A}^n } \quad ( n = 1,2,3 , \cdots ) ,$$
$\left\{ b _ { n } \right\}$ satisfies
$$b _ { 1 } = \mathbf { B } , \quad b _ { n + 1 } - \mathbf { C } \; b _ { n } + \mathbf { D } = 0 \quad ( n = 1,2,3 , \cdots ) .$$
This recurrence formula can be transformed into
$$b _ { n + 1 } - \mathbf { E } = \mathbf{F} ( b_n - \mathbf{E} )$$
Next, when we define a sequence $\left\{ c _ { n } \right\}$ by
$$c _ { n } = b _ { n } - \mathbf { E } \quad ( n = 1,2,3 , \cdots ) ,$$
$\left\{ c _ { n } \right\}$ is a geometric progression such that the first term is $\mathbf{G}$ and the common ratio is $\mathbf{H}$.
Hence we have
$$a _ { n } = \mathbf { I } ^ { n } \left( \mathbf { J } \cdot \mathbf { K } ^ { n - 1 } + \mathbf { L } \right) \quad ( n = 1,2,3 , \cdots ) .$$