I'm a beginner who is starting to learn about linear recurrence relations. I've just come across a problem where I'm not sure how to progress.
Going off of my notes linear recurrence was solved using this form
$a_{n}= C_{1}a_{n-1}+C_{2}a_{n-2}$
where the C's are constants
that form is then manipulated and used with the quadratic formula … etc..
My problem is since this method uses $a_{n-1}$ and $a_{n-2}$, I'm not sure how to solve a linear recurrence problem for a formula that doesn't have those.
For reference here is the problem
$a_{n} = 2^{n}(1 + (−2)^{n})$
What is the method used to solve these types of recurrence problems?
After some clarification I realized that the problem is asking for me to find the linear recurrence of the form $a_{n}= C_{1}a_{n-1}+C_{2}a_{n-2}$, but I'm still not sure how to do this
Best Answer
As you have stated $$a_{n} = 2^{n}(1 + (−2)^{n})=2^n+(-4)^n$$ So the recurrence relation must have an auxiliary equation given by $$(r-2)(r+4)=0$$ in order to have the solutions $r=2,-4$. Expanding this gives $$r^2+2r-8=0$$ $$r^2=-2r+8$$ Hence the original recurrence relation is $$a_n=-2a_{n-1}+8a_{n-2}$$