Difference between revisions of "Fibonacci numbers"
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The Fibonacci sequence is defined by | The Fibonacci sequence is defined by | ||
$$F_{n+2}=F_n+F_{n+1},F_1=F_2=1.$$ | $$F_{n+2}=F_n+F_{n+1},F_1=F_2=1.$$ | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The following series holds and converges for all $|x| \leq \dfrac{1}{\phi}$, where $\phi$ denotes the [[golden ratio]]: | ||
| + | $$\dfrac{x}{1-x-x^2} = \displaystyle\sum_{k=1}^{\infty} F_k x^k.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> proof goes here █ | ||
| + | </div> | ||
| + | </div> | ||
=Videos= | =Videos= | ||
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=External links= | =External links= | ||
| − | [http://www.fq.math.ca/ The Fibonacci Quarterly] | + | [http://www.fq.math.ca/ The Fibonacci Quarterly]<br /> |
| + | [http://matheducators.stackexchange.com/questions/2021/what-interesting-properties-of-the-fibonacci-sequence-can-i-share-when-introduci]<br /> | ||
Revision as of 03:14, 2 December 2015
The Fibonacci sequence is defined by $$F_{n+2}=F_n+F_{n+1},F_1=F_2=1.$$
Contents
Properties
Theorem: The following series holds and converges for all $|x| \leq \dfrac{1}{\phi}$, where $\phi$ denotes the golden ratio: $$\dfrac{x}{1-x-x^2} = \displaystyle\sum_{k=1}^{\infty} F_k x^k.$$
Proof: proof goes here █
Videos
The Golden Ratio & Fibonacci Numbers: Fact versus Fiction
Doodling in Math: Spirals, Fibonacci, and Being a Plant (1 of 3)
Fibonacci mystery
See also
Golden ratio
Reciprocal Fibonacci constant
Lucas numbers
