A proof about the fibonnaci numbers (simple for you guys)

In summary, the conversation discusses the problem of proving that F_1*F_2+F_2*F_3+...+F_{2n-1}*F_{2n}=F^2_{2n}. It is mentioned that an earlier proof by induction was used to show that F_{2n}=F_1+F_2+...+F_{2n-1}, but it is not necessary to use in the current proof. The correct way to approach the proof by induction is explained, and it is advised to assume the result is true for n and try to prove it for n+1. It is also noted that adding the last term with n+1 substituted for n to the sum does not
  • #1
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The problem is as stated:
Prove that \(\displaystyle F_1*F_2+F_2*F_3+...+F_{2n-1}*F_{2n}=F^2_{2n}\)

But earlier in my text I proved by induction that \(\displaystyle F_{2n}=F_1+F_2+...+F_{2n-1}\). Do I need to use this earlier proof in my current proof. I tried adding \(\displaystyle F_{2n+1}F_{2n+2}\) to the right and left hand side of the first equation and tried to find \(\displaystyle F_{2n+1}F_{2n+2}+F^2_{2n}=F^2_{2n+2}\) but that doesn't seem to be going anywhere. (Why doesn't that seem to work in this case? Because I am multiplying two sums together?)

Am I wrong in assuming that I am supposed to prove this by induction?
 
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  • #2
Your equation \(\displaystyle F_{2n}=F_1+F_2+...+F_{2n-1}\) must have a typo, because it's incorrect (try it for \(\displaystyle n = 2\), say). However, I don't think you need it to solve the given problem. Proving by induction is correct, but your induction step isn't set up quite right. Assume the result is true for n and try to prove it for n + 1. Thus, we are trying to prove that

\(\displaystyle F_1F_2 + F_2F_3 + \cdots + F_{2n-1}F_{2n} + F_{2n}F_{2n+1} + F_{2n+1}F_{2n+2} = F_{2n+2}^2\)

(because \(\displaystyle 2(n+1) = 2n +2\)), and we know by induction that

\(\displaystyle F_1F_2 + F_2F_3 + \cdots + F_{2n-1}F_{2n} = F_{2n}^2\)

Can you take it from there?
 
  • #3
Yea, after you told me how to set it up it took about ten seconds :P, and here I had sat and wondered about it for like an hour. I had assumed that I could plug N+1 into \(\displaystyle F_{2n-1}F_{2n}\) and add that back to the left hand side of the equation and get what equaled \(\displaystyle F^2_{2n+2}\). So now I know that adding the last term with n+1 substituted for n to the sum doesn't necessarily result in the actual n+1 sum as a whole. (Also thanks to you I solved the next three problems I couldn't solve :D)
 
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FAQ: A proof about the fibonnaci numbers (simple for you guys)

What are the Fibonacci numbers and why are they important in mathematics?

The Fibonacci numbers are a sequence of numbers where each number is the sum of the two preceding ones, starting from 0 and 1. They are important in mathematics because they have numerous applications in various fields such as number theory, geometry, and even in nature.

What is the proof about the Fibonacci numbers?

The proof about the Fibonacci numbers is a mathematical demonstration of why the sequence follows a specific pattern and how it can be calculated. It shows that the sequence can be derived using a simple formula and that it has many interesting properties.

Is the proof about the Fibonacci numbers difficult to understand?

The proof about the Fibonacci numbers may seem complex at first, but it can be broken down into simpler steps and explained in a way that is easy to understand. It may require some basic knowledge of algebra and number theory, but it is not overly complicated.

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Are there any limitations or exceptions to the proof about the Fibonacci numbers?

While the proof about the Fibonacci numbers is generally accepted and widely used, there are some limitations and exceptions to its application. For example, in some cases, the sequence may not follow the expected pattern due to external factors or deviations from the formula used to calculate it.

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