Prove this fibonacci sequence w/o binet formula

In summary, the conversation is about proving the equation $u_{m+n} = u_{m-1}u_n + u_mu_{n+1}$ for all natural numbers $m$ and $n$, without using the Binet formula. The person is unsure of where to start and asks if they need to include the derivation of the Binet formula in the proof. They also mention having previous experience with proofs involving the Fibonacci numbers, but not with two unknowns.
  • #1
skate_nerd
176
0
I have a problem and honestly have no idea even where to start. I've been staring at it and thinking about it for over 24 hours...
Let \(u_n\) denote the \(n^{th}\) Fibonacci number. Without using the Binet formula for \(u_n\), prove the following for all natural numbers \(m\) and \(n\) with \(m\geq{2}\):
$$u_{m+n}=u_{m-1}u_n+u_mu_{n+1}$$

I have gone through a couple proofs regarding the Fibonacci numbers in this class before, but never one with two unknowns, and also never one where we weren't able to use the Binet formula. Without the Binet formula, what do I have to work off of? Do I need to write the derivation for the Binet formula within this whole proof?
 
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  • #2
Try induction on $n$.
 

FAQ: Prove this fibonacci sequence w/o binet formula

What is the Fibonacci sequence?

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding numbers. It starts with 0 and 1, and each subsequent number is the sum of the two numbers before it. The sequence looks like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

What is the Binet formula for the Fibonacci sequence?

The Binet formula is a mathematical formula that can be used to find any term in the Fibonacci sequence. It is written as:
Fn = [Phi^n - (-Phi)^(-n)] / √5
Where Phi is the golden ratio, approximately equal to 1.618.

Why is it important to prove the Fibonacci sequence without using the Binet formula?

Proving the Fibonacci sequence without using the Binet formula helps to solidify our understanding of the sequence and its properties. It also showcases the various mathematical techniques and principles that can be used to solve problems, rather than relying on a single formula.

How can the Fibonacci sequence be proven without using the Binet formula?

There are multiple ways to prove the Fibonacci sequence without using the Binet formula. One approach is to use mathematical induction, where we prove that the formula holds true for the first few terms and then show how it can be extended to the next term. Another approach is to use the relationship between the Fibonacci sequence and the golden ratio to derive a proof.

What are some real-life applications of the Fibonacci sequence?

The Fibonacci sequence can be found in nature, such as in the branching patterns of trees or in the arrangement of seeds in a sunflower. It is also used in various fields of science and mathematics, such as in computer algorithms, financial analysis, and in the study of population growth. Additionally, the sequence has also been used in art and music, as it has a aesthetically pleasing pattern.

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