Difference between revisions of "Fibonacci sequence"
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== Binet's formula == | == Binet's formula == | ||
+ | {{main|Binet's formula}} | ||
− | + | Binet's formula is an explicit formula used to find the <math>n</math>th term of the Fibonacci sequence. | |
It is <math>\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right)</math>. | It is <math>\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right)</math>. | ||
− | + | It is so named because it was derived by mathematician Jacques Philippe Marie Binet, though it was already known by Abraham de Moivre. | |
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==Problems== | ==Problems== |
Revision as of 21:03, 7 December 2007
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The Fibonacci sequence is a sequence of integers in which the first and second term are both equal to 1, and each subsequent term is the sum of the two preceding it. Often, there is a zero-th term added in equal to 0. The first few terms are
.
The Fibonacci sequence can be written recursively as . There is also an explicit definition below.
Contents
[hide]Phi
Ratios between successive terms, , , , , , tend towards the limit phi.
Binet's formula
- Main article: Binet's formula
Binet's formula is an explicit formula used to find the th term of the Fibonacci sequence. It is .
It is so named because it was derived by mathematician Jacques Philippe Marie Binet, though it was already known by Abraham de Moivre.
Problems
Introductory
- The Fibonacci sequence starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?
- Find .
Intermediate
- Seven line segments, with lengths no greater than 10 inches, and no shorter than 1 inch, are given. Show that one can choose three of them to represent the sides of a triangle. (Manhattan Mathematical Olympiad 2004)
- Except for the first two terms, each term of the sequence is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encounted. What positive integer produces a sequence of maximum length?
- A fair coin is to be tossed times. Let , in lowest terms, be the probability that heads never occur on consecutive tosses. Find .
- Find if and are integers such that is a factor of .
Olympiad
- Determine the maximum value of , where and are integers satisfying and .
See also
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