Identifying R1 and R2 when solving Recursion relations

Question

We are learning to solve recursion relations. When I get this step, does it matter if I define $r_1$ as 5 or 2 in this example?

Answer 1

It makes no difference. By the way, you want $r^k=7r^{k-1}-10r^{k-2}$. But your solutions are fine.

If you use $r_1=5$, $r_2=2$, you will be finding $A$ and $B$ such that $A\cdot5^k+B\cdot 2^k$ satisfies the initial conditions. If you use $r_1=2$, $r=5$, you will be finding $A$ and $B$ such that $A\cdot2^k+B\cdot 5^k$ satisfies the initial conditions. Ultimately you get equivalent general formulas.

There is good reason to use $r_1=5$, $r_2=2$, since the $5^k$ term is the dominant term.