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Showing a Sequence Also Satisfies the Fibonacci Sequence

  • Thread starter Illania
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  • #1
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Homework Statement



The problem:
Let r satisfy r2= r + 1. Show that the sequence an = Arn, where A is constant, satisfies the Fibonacci sequence an = an-1 + an-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 = Arn-1 + Arn-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(rn-1 + rn-2), and then say that rn = rn-1 + n-2, but I have no idea what to do from here.
 

Answers and Replies

  • #2
Dick
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Homework Helper
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Homework Statement



The problem:
Let r satisfy r2= r + 1. Show that the sequence an = Arn, where A is constant, satisfies the Fibonacci sequence an = an-1 + an-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 = Arn-1 + Arn-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(rn-1 + rn-2), and then say that rn = rn-1 + n-2, but I have no idea what to do from here.
Divide both sides by r^(n-2).
 
  • #3
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So, I'm guessing I can manipulate the exponents as you would normal ones, and dividing both sides by rn-2 would result in r2 = rn-1, correct?
 
  • #4
Dick
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So, I'm guessing I can manipulate the exponents as you would normal ones, and dividing both sides by rn-2 would result in r2 = rn-1, correct?
No? What are r^n/r^(n-2), r^(n-1)/r^(n-2) and r^(n-2)/r^(n-2)??
 
  • #5
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rn/rn-2 = rn-1/rn-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.
 
Last edited:
  • #6
Dick
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rn/rn-2 = rn-1/rn-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).
 
  • #7
26
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Thank you for the explanation! I was thinking in just one way rather than looking at the problem from multiple angles.
 

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