Find unique solutions to recurrence relation $a_n =10a_{n-1}-32a_{n-2}+32a_{n-3}$ with $a_0=5$, $a_1=18$, $a_2=76$

Question

How to find the unique solutions to the recurrence relation given initial conditions and using the characteristic root technique?

$a_n =10a_{n-1}-32a_{n-2}+32a_{n-3}$ with $a_0=5$, $a_1=18$, $a_2=76$

Using the characteristic root technique, I create the characteristic root equation $x^3 -10x^2+32x-32=0$, which reduces to $(x-4)^2(x-2)=0$, so the characteristic roots are x=4 and x=2

I am confused here about how to set up my system of equations because for differing roots we use $a_n=ar_1^n + br_2^n$ where $r$ are the roots. While for repeated roots we use $a_n=ar^n + bnr^n$. How do I set up the system of equations to solve when an equation has repeated and differing roots?

Answer 1

Hint

The treatment with the repeated roots is as follows:

if a root $r$ was repeated of order, say $k$ , then the corresponding terms in the general recurrence will become$$t_n=P_k(n)\cdot r^n$$where $P_k(n)$ is a polynomial of degree $k-1$.

In our case, it will be:$$a_n=(a+bn)\cdot 4^n+c\cdot 2^n$$