Q:

Finding a closed form expression for the following recurrence relation.

Given that:
$S_0  =  0$
$S_1  =  3$
$S_n  =  S_{n-1}  +  6S_{n-2}$          for $n ≥ 2$
What are the steps I would take to find the closed form expression for the following recurrence relation?

A:

What you are being asked for is an equation that has $x_n$ on the left side and a formula in $n$ on the right side containing just familiar functions like polynomials and exponentials and only finitely many of them. 
First, write it like this for simplicity: $S_0=0$, $S_1=3$, $S_n=S_{n-1}+6S_{n-2}$. 
Now, let's suppose there is a solution of the form $S_n=x^n$ for some $x$. 
Then the equation says $x^n=x^{n-1}+6x^{n-2}$, simplifies to $x^2-x-6=0$, 
which undoubtedly you would be able to solve. 
You'll get two values of $x$ that work, let's call them $x_1$ and $x_2$, and then anything of the form $$\alpha x_1^n+\beta x_2^n$$ will work also, for any numbers $\alpha$ and $\beta$. 

