Showing a Sequence Also Satisfies the Fibonacci Sequence

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In summary, the problem is to show that the sequence an = Arn, where A is constant, satisfies the Fibonacci sequence an = an-1 + an-2 for n > 2 by manipulating the given equations and showing that they are equal. The relevant equations are r^2 = r + 1 and an = Arn, and the attempt at a solution involves manipulating the exponents and dividing both sides by r^(n-2).
  • #1
Illania
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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.
 
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  • #2
Illania said:

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
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
Illania said:
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
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.
 
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  • #6
Illania said:
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
Thank you for the explanation! I was thinking in just one way rather than looking at the problem from multiple angles.
 

FAQ: Showing a Sequence Also Satisfies the Fibonacci Sequence

What is the Fibonacci Sequence?

The Fibonacci Sequence is a mathematical sequence in which each number is the sum of the two preceding numbers. It starts with 0 and 1, and the next number is found by adding the previous two numbers together. The sequence continues infinitely, and the first few numbers are 0, 1, 1, 2, 3, 5, 8, and so on.

What does it mean for a sequence to satisfy the Fibonacci Sequence?

In order for a sequence to satisfy the Fibonacci Sequence, each number in the sequence must be the sum of the two preceding numbers, just like in the original Fibonacci Sequence. This means that the sequence must follow the same pattern of increasing by adding the two previous numbers together.

How can you show that a sequence satisfies the Fibonacci Sequence?

To show that a sequence satisfies the Fibonacci Sequence, you can use mathematical induction. This involves proving that the first two numbers in the sequence are 0 and 1, and then showing that each subsequent number is the sum of the two preceding numbers. If this pattern holds true for every number in the sequence, then it satisfies the Fibonacci Sequence.

Why is it important to show that a sequence satisfies the Fibonacci Sequence?

Showing that a sequence satisfies the Fibonacci Sequence can help to identify patterns and relationships within the sequence. It can also provide evidence for the presence of the Golden Ratio, a mathematical ratio that is closely related to the Fibonacci Sequence and is found in many natural phenomena.

What are some real-world applications of the Fibonacci Sequence?

The Fibonacci Sequence has many real-world applications, including in fields such as architecture, music, and computer science. It can be seen in the branching of trees, the arrangement of leaves on a stem, and the spirals of shells and galaxies. In computer science, the sequence is used in algorithms for searching and sorting data. In music, the sequence is often used to create pleasing rhythms and melodies.

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