I've never felt dumber: me understand Fibonacci identity.

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In summary, the conversation discusses the sequence E(n) and its relation to the Fibonacci sequence. It is explained that E(n) can be represented as F(n-1)A+F(n)B and that it is essentially the Fibonacci sequence shifted to the left. The identity F(a+b+1)=F(a+1)F(b+1)+F(a)F(b) is also introduced, causing confusion and frustration for the speaker. Eventually, it is clarified that this identity can be derived from the previous representations of E(n).
  • #1
moe darklight
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This is crazy. I have no idea what the textbook is saying at the end.

if we have E(0)=A, E(1)=B, And E(n)=E(n-1)+E(n-2); then E(n)=F(n-1)A+F(n)B

(Where F(n) denotes a Fibonacci number.)

So far, so good. Then this flies at me out of nowhere:

We can start with two consecutive Fibonacci numbers A=F(a) and B=F(a+1). Then the sequence E(n) is just the Fibonacci sequence shifted to the left (agreed). Hence we get the following identity:

F(a+b+1)=F(a+1)F(b+1)+F(a)F(b)

We do?? Where the hell did that come from?

I've never stared at something for so long without having the slightest clue what is going on. I held up the whole class today for like 20 minutes, and it seemed to make sense when the prof explained it . . . but now I look at it again, and whatever I learned, I un-leaned. Ugh, I'm so frustrated.
 
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  • #2
E(n)=F(n-1)A+F(n)B

E(n)=F(n-1)E(0)+F(n)E(1)

E(n+1)=F(n)E(0)+F(n+1)E(1)

and since E(n)=F(b+n) (by definition),

F(b+n+1)=F(n)F(b)+F(n+1)F(b+1)
 
  • #3
:rolleyes: of course. Thanks.
 

1. What is the Fibonacci identity?

The Fibonacci identity is a mathematical equation that states that the sum of the squares of the first n Fibonacci numbers is equal to the product of the nth and (n+1)th Fibonacci numbers.

2. Why is it important to understand the Fibonacci identity?

The Fibonacci identity is important because it has various applications in mathematics, computer science, and nature. It can be used to solve problems in number theory, combinatorics, and algorithms, and it can also be seen in the growth patterns of plants and animals.

3. How can I use the Fibonacci identity?

The Fibonacci identity can be used to solve problems involving Fibonacci numbers, such as finding the sum of the squares of a certain number of Fibonacci numbers or proving other mathematical identities. It can also be used in real-world applications, such as predicting population growth or analyzing financial markets.

4. Is there a real-life example of the Fibonacci identity?

Yes, one example is the spiral shape found in the center of a sunflower. The seeds in this spiral follow the pattern of the Fibonacci sequence, and the number of spirals in each direction follows the Fibonacci numbers. This is an example of how the Fibonacci identity can be observed in nature.

5. How can I improve my understanding of the Fibonacci identity?

One way to improve your understanding of the Fibonacci identity is to practice solving problems involving Fibonacci numbers and the identity. You can also read about its various applications and see how it is used in different fields. Additionally, discussing the concept with others and seeking out additional resources, such as textbooks or online tutorials, can also help improve your understanding.

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