Question

Suppose a recurrence relation

an=7an−1−12an−2

where a1=16 and a2=52

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.

**If the explicit formula is in the form of Formula 2, consider p < q.

Determine

p and q

Answer 1

 characteristic equation:

$x^2-7x+12=0$

$x=\frac{7\pm\sqrt{49-48}}{2}$

$x_1=3,x_2=4$

solution:

$a_n=p\cdot 3^n+q\cdot 4^n$

$a_1=3p+4q=16$

$a_2=9p+16q=52$

$4q=4 \implies q=1$

$p=(16-4q)/3=4$