Essay

$$\text { Let } a _ { 1 } = 2 , a _ { 2 } = 9 \text {, and } a _ { n } = 2 a _ { n - 1 } + 3 a _ { n - 2 } \text { for } n \geq 3 \text {. Show that } a _ { n } \leq 3 ^ { n } \text { for all positive integers } n \text {. }$$