I've read some ways about how to derive a formula from a recursive definition, but what about this one?
I started solving this formula
$$ a_n = 2^n + 5^n n , n \in \mathbb{N} $$
gives you the recursive definition of what and how you go about figuring that out? Any tips for me?
My calculations so far
$$ (r-2)(r-5) = r^2 - 7r + 10 $$
Am i doing right?
Because of the coefficient $n$ here, you'll need a double root at $r=5$. Thus you should look at the characteristic polynomial $$(r-2)(r-5)^2=r^3-12r^2+45r-50$$
It follows that the recursion we want is $$a_n=12a_{n-1}-45a_{n-2}+50a_{n-3}$$ with initial conditions $$a_0=1\quad a_1=7\quad a_2=54$$