Exploring Recursive Lucas & Fibonacci Sequences: Can You Prove It?

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In summary, the conversation discusses recursive definitions of Lucas and Fibonacci sequences and a relationship between the sums of these sequences. The speaker asks for a counterexample using an inductive approach and mentions that the formula can be proven through induction. They also mention a similar relation that is independent of the starting values.
  • #1
ghostskwid
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Hi I am playing around with recursive definitions of Lucas and Fibonacci sequences:

I came across a relationship

L0 + L1 + L2 + L3 ... Ln = sum(i = 0, n) Li = Ln+2 -1;

Sorry for the horrible notation, but could anyone provide a counter example using an inductive approach? I get the counter example through guessing, but am having a hard time proving it definitively.
 
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  • #2
Any counterexample would show that your formula is wrong, it does not matter how you got that counterexample.

You can show this formula via induction, this is an easy example of induction.
Actually, there should be a similar relation independent of the starting values, where just the constant in the formula has to be changed.
 

Related to Exploring Recursive Lucas & Fibonacci Sequences: Can You Prove It?

1. What is a recursive Lucas sequence?

A recursive Lucas sequence is a series of numbers where each subsequent number is the sum of the two preceding numbers. The first two numbers in the sequence are typically defined as 2 and 1. So, the sequence starts as 2, 1, 3, 4, 7, 11, and so on.

2. How is a recursive Lucas sequence different from a Fibonacci sequence?

A recursive Lucas sequence is similar to a Fibonacci sequence in that each subsequent number is the sum of the two preceding numbers. However, the first two numbers in a Fibonacci sequence are typically defined as 0 and 1, while the first two numbers in a recursive Lucas sequence are defined as 2 and 1. This leads to slight differences in the sequences as they progress.

3. Can you provide an example of a recursive Lucas sequence?

As stated before, a recursive Lucas sequence starts with the numbers 2 and 1. So, the first few numbers in the sequence are 2, 1, 3, 4, 7, 11, 18, 29, and so on.

4. How can you prove that a recursive Lucas sequence will continue infinitely?

A recursive Lucas sequence will continue infinitely because the definition of the sequence is recursive, meaning that each number is determined by the previous two numbers. As long as we have a starting point of 2 and 1, the sequence will continue to generate new numbers.

5. What are some real-world applications of recursive Lucas and Fibonacci sequences?

Recursive Lucas and Fibonacci sequences can be found in nature, such as in the branching patterns of trees and the spirals of seashells. They are also used in computer algorithms, financial modeling, and cryptography.

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