Which recurrence relation is greater?

[swtlilsoni]

Homework Statement

A_n+2=6A_n+1+6A_n
A₁=A₂=1
B_n+2=5B_n+1+9B_n
B₁=B₂=10¹⁰⁰

a) is it true that B_n>A_n for every integer n > 0?
b) is it true that B_n>A_n for infinetly many integers n>0?

The Attempt at a Solution

It just seems like these are increasing functions since they both start with positive integers and are only addition. Thus the function with B has a greater initial value so it should be greater.
The only way this wouldn't be true is if the rate at which A is increasing is greater than the rate at which B is increasing.
However I do not know how to find the rate.

 

[CompuChip]

Probably there is some nice trick for this, but personally I would try solving for the direct formula (for inspiration, check wikipedia)

 

[swtlilsoni]

Here is what my professor wrote (I don't completely understand what he did):
A_n= c₁[tex]\alpha[/tex]ⁿ + c₂[tex]\beta[/tex]ⁿ
c₁,c₂= 3 [tex]\pm[/tex] [tex]\sqrt{}15[/tex]
B_n= c₃[tex]\delta[/tex]ⁿ + c₄[tex]\varpi[/tex]ⁿ
c₃,c₄= (5 [tex]\pm[/tex] [tex]\sqrt{}61[/tex])/2

3+[tex]\sqrt{}15[/tex]=[tex]\alpha[/tex]=6.8
3-[tex]\sqrt{}15[/tex]=.9ⁿ = approaching zero
(5 + [tex]\sqrt{}61[/tex])/2 = [tex]\delta[/tex] = 6.5
(5 - [tex]\sqrt{}61[/tex])/2 = [tex]\varpi[/tex] = -1.5

A_n > B_n for all n
since [tex]\alpha[/tex] > [tex]\delta[/tex] > [tex]\varpi[/tex]

I have NO CLUE how he got this. It is possible I could have copied some of it wrong, but this is the general idea of what he did. I'm sure it could be helpful in figuring out how this problem should be done?