I'm currently being introduced to $2^{nd}$ order linear homogenous recurrence relations for the first time. I was working through a first example in my textbook and came into some trouble. Here is the section of my textbook:
What is confusing me is how they write $a_n = c_1(2^n) + c_2(-3)^n$ as the general solution near the end of the second paragraph. Why do we add our two solutions when it is clear $a_n = 6a_{n-2} - a_{n-1}$ from what we were given in the first line?
We want a formula for $a_n$ that doesn't involve other $a$'s, and thus allows us (once we know the constants $c_1$ and $c_2$ to calculate any $a_n$ just from $n$.