1
$\begingroup$

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?

$\endgroup$
3
$\begingroup$

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:$$a_n=(a+bn)\cdot 4^n+c\cdot 2^n$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.