Answer to Question #271551 in Discrete Mathematics for nur

Question

Suppose a recurrence relation

an=4an−1−4an−2

where a1=14 and a2=40

can be represented in explicit formula, either as:

Formula 1:

an=pxn+qnxn

or

Formula 2:

an=pxn+qyn

where

x

and

y

are roots of the characteristic equation.

Determine

p

and

q

.

Answer 1

$$a_n-4a_{n−1}+4a_{n−2}=0$$

characteristic equation:

$$r^2-4r+4=0$$

$$(r-2)^2=0$$

$r=2$ is the only characteristic root. Therefore we know that the solution to the recurrence relation has the form

$$a_n=a2^n+bn2^n$$

Given $a_1=14 ,a_2=40$

$$a_1=a(2^1)+b(1)(2^1)=14$$

$$a_2=a(2^2)+b(2)(2^2)=40$$

$$a+b=7$$ $$\\ a+2b=10$$

$$a=4$$ $$b=3$$

Therefore the solution to the recurrence relation is

$$a_n=4\cdot2^n+3n\cdot2^n$$