Showing a Sequence Also Satisfies the Fibonacci Sequence

[Illania]

Homework Statement

The problem:
Let r satisfy r²= r + 1. Show that the sequence a_n = Arⁿ, where A is constant, satisfies the Fibonacci sequence a_n = a_n-1 + a_n-2 for n > 2.

Homework Equations

The given equations above are the only relevant equations.

The Attempt at a Solution

I think have to show that Arⁿ = Ar^n-1 + Ar^n-2, but I'm not sure how to manipulate the given equations to achieve such a thing. I can factor one side so that Arⁿ = A(r^n-1 + r^n-2), and then say that rⁿ = r^n-1 + ^n-2, but I have no idea what to do from here.

 

[Dick]

Homework Statement

The problem:
Let r satisfy r²= r + 1. Show that the sequence a_n = Arⁿ, where A is constant, satisfies the Fibonacci sequence a_n = a_n-1 + a_n-2 for n > 2.

Homework Equations

The given equations above are the only relevant equations.

The Attempt at a Solution

I think have to show that Arⁿ = Ar^n-1 + Ar^n-2, but I'm not sure how to manipulate the given equations to achieve such a thing. I can factor one side so that Arⁿ = A(r^n-1 + r^n-2), and then say that rⁿ = r^n-1 + ^n-2, but I have no idea what to do from here.

Divide both sides by r^(n-2).

 

[Illania]

So, I'm guessing I can manipulate the exponents as you would normal ones, and dividing both sides by r^n-2 would result in r² = r^n-1, correct?

 

[Dick]

So, I'm guessing I can manipulate the exponents as you would normal ones, and dividing both sides by r^n-2 would result in r² = r^n-1, correct?

No? What are r^n/r^(n-2), r^(n-1)/r^(n-2) and r^(n-2)/r^(n-2)??

 

[Illania]

rⁿ/r^n-2 = r^n-1/r^n-2 + 1, which simplifies to r² = r + 1.

I'm still unsure of how this can be used to prove that Arⁿ also satisfies the Fibonacci sequence.

EDIT: Does showing that the two set equal to each other simplifies to r² = r + 1 prove that the two are equal? If so, why is that? This question really seems to be going right over my head.

 

[Dick]

rⁿ/r^n-2 = r^n-1/r^n-2 + 1, which simplifies to r² = r + 1.

I'm still unsure of how this can be used to prove that Arⁿ also satisfies the Fibonacci sequence.

EDIT: Does showing that the two set equal to each other simplifies to r² = r + 1 prove that the two are equal? If so, why is that? This question really seems to be going right over my head.

It doesn't prove the two are equal. It proves that if r^2=r+1 then Ar^n=Ar^(n-1)+Ar^(n-2). All of your steps are reversible. Multiply both sides of r^2=r+1 by Ar^(n-2).

 

[Illania]

Thank you for the explanation! I was thinking in just one way rather than looking at the problem from multiple angles.