S$($n$)=3$S$($n$-1)+2$

S$(1)=2$

$($a$)$ Draw a recursion tree for S$(n).$ Include the recursive terms for the nodes in a few levels, and also show what the total non$-$recursive term amounts to for each of those levels. State what the depth of the tree is$,$ and explain how you determined this. Provide a convincing argument that S$(n)=(g(n)),$ for some simple function $g(n).$

$($b$)$ Use induction to prove that S$(n)=O(g(n)),$ for the same $g(n)$ as in part $($a$).$ Use the same style and show the same level of detail as in the course notes.