Fibonacci recurance relation

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In summary, the conversation involves finding a recurrence relation for expressing both f_{n+1} and f_{n} in terms of f_{n+2} and f_{n+3}. The proposed solution is f_{n+1}+f_{n+2}+f_{n+3}...f_{n} and there are other ways of solving it mentioned in the book, but they do not use f_{n+3}. The conversation also includes a discussion about formatting equations in LaTeX.
  • #1
morbello
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i have to find the recurrence relation to express both f[tex]_{n+1}[/tex] and f[tex]_{n}[/tex] with f[tex]_{n+2}[/tex] and f[tex]_{n+3}[/tex]

my answer is
f[tex]_{n+1}[/tex]+f[tex]_{n+2}[/tex]+f[tex]_{n+3}[/tex]...f[tex]_{n}[/tex]

ive got other ways off doing it in the book but they do not use f[tex]_{n+3}[/tex]

these are

f[tex]_{n+2}[/tex]+f[tex]_{f-1}[/tex]-f[tex]_{n}[/tex]


they all should be in lower case but i can not seam to get the post to do it i hope it does not make it hard to help me with the question.



The Attempt at a Solution






 
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  • #2
hi morbello

you can click on any latex code to see how it is typed, easiest to put tex quotes around your whole equation


but I'm not too sure what you're trying to do... ;) shouldn't you have an equals sign somewhere?

something like
[tex] f^{n+1} = f(n+1) = f(n) + ...?[/tex]
 
  • #3
:

The Fibonacci recurrence relation is a mathematical formula that describes the relationship between consecutive terms in the Fibonacci sequence. This sequence is defined as a series of numbers where each number is the sum of the two preceding numbers, starting with 0 and 1. The recurrence relation for this sequence is f_{n+2} = f_{n+1} + f_{n}.

To express both f_{n+1} and f_{n} with f_{n+2} and f_{n+3}, we can use the relation f_{n+3} = f_{n+2} + f_{n+1}. This means that f_{n+1} = f_{n+3} - f_{n+2} and f_{n} = f_{n+2} - f_{n+1}. Substituting these expressions into the original recurrence relation, we get f_{n+2} = (f_{n+3} - f_{n+2}) + (f_{n+2} - f_{n+1}). Simplifying this equation, we get f_{n+3} = 2f_{n+2} - f_{n+1}.

Therefore, the recurrence relation to express both f_{n+1} and f_{n} with f_{n+2} and f_{n+3} is f_{n+3} = 2f_{n+2} - f_{n+1}. This relation can be used to find any term in the Fibonacci sequence by plugging in the appropriate values for n.
 

1. What is the Fibonacci recurrence relation?

The Fibonacci recurrence relation is a mathematical formula that defines the sequence of numbers known as the Fibonacci sequence. The formula states that each number in the sequence is the sum of the two preceding numbers, starting with 0 and 1. The sequence goes as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc.

2. Who discovered the Fibonacci recurrence relation?

The Fibonacci recurrence relation was first described by the Italian mathematician Leonardo Fibonacci in his book Liber Abaci in 1202. However, the sequence itself was known to Indian mathematicians as early as 200 BC.

3. What are some real-world applications of the Fibonacci recurrence relation?

The Fibonacci recurrence relation has numerous applications in various fields, including mathematics, biology, finance, and art. In mathematics, it can be used to model population growth, and in biology, it can help in understanding the branching patterns of trees and other natural phenomena. In finance, the sequence is used in technical analysis to predict market trends. In art, the Fibonacci sequence is often used to create visually appealing compositions.

4. Can the Fibonacci recurrence relation be generalized to other sequences?

Yes, the Fibonacci recurrence relation can be generalized to other sequences by changing the initial values and the rule for combining the numbers. For example, the Lucas sequence follows the same rule as the Fibonacci sequence, but with different initial values of 2 and 1.

5. How is the Fibonacci recurrence relation related to the golden ratio?

The golden ratio, also known as the divine proportion, is a mathematical concept that is closely related to the Fibonacci sequence. As the numbers in the Fibonacci sequence get larger, the ratio between consecutive numbers approaches the golden ratio, which is approximately 1.618. This ratio can be seen in many natural and man-made structures, such as the spiral of a seashell and the design of the Parthenon in Greece.

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