Fibonacci Series

In 1202, Leonardo of Pisa (filius Bonacci) published Liber Abaci, or ABAC book. At that time, the book has not attracted attention, because it used arabic numbers.
However, a problem of that book remained famous:

iepure.jpg "A certain man put a pair of rabbits in a place surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?"
fib.gif

Leonardo established law of multiplication of a pair of house rabbits, that is a mathematical expression corresponding to organic growth, and this proved to have very broad applications. Analyzed the problem's solution is obtained Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21 … In mathematical terms:

Numn = Numn-1 + Numn-2

It may be said that the Fibonacci string is a double additive.


The ratio of two consecutive terms of Fibonacci's string value represents an approximate of golden number, Ø, and the limit of this report is Ø.
Ø2=Ø+1
The question may arise whether a positive integer z is a Fibonacci number. Since Fn is the closest integer to $\Phi^n/\sqrt{5}$, the most straightforward, brute-force test is the identity: $F\bigg(\bigg\lfloor\log_\Phi(\sqrt{5}z)+\frac{1}{2}\bigg\rfloor\bigg)=z,$
which is true if and only if z is a Fibonacci number.
Alternatively, a positive integer $z$ is a Fibonacci number if and only if one of $5z^2+4$ or $5z^2-4$ is a perfect square.

Continued fraction representation for the golden ratio:

(1)
\begin{align} \Phi=1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{\;\;\ddots\,}}} \;. \end{align}

We affirm that our number Phi is closely linked to the row of
Fibonacci. For those who do not know of Fibonacci series is defined
by f [0] = 0 f [1] = 1 f [n] = f [0] + f [1] (each n32).
This series cast (in a naive way) increase the population of rabbits.
It is assumed that the rabbits were put in pairs once every month after
reach the age of two months. Also, chickens do not die and never
are one male and one female.
Thus, the number of existing pairs of rabbits after n months would
have to be f - [n]. We put the question and may have in common with F
Fibonacci's series? This is a remarkable idea of mathematics.
To begin to see that F is a fraction to look infinitã.Acum partial fractions:
All the fractions are reports of Fibonacci numbers
Successive which "motivated" theorem that says that:
In words we can say that as the n approaching infinity,
terms of the n +1- th and the n-th row of the Fibonacci
approaching Theorems of F. This is valid for any sequence
Arbitrary recurring satisfying f [n] = f [0] + f [1] (each n32), with the property that the first two terms are different.
Fascination of Fibonacci numbers
Fibonacci sequence of numbers of fascinated by the history along the many scientists, mathematicians, physicists, biologists, and continue to do so even today.

History
Fibonacci (1170 - 1240) is regarded as one of the greatest European mathematicians of the Middle Ages. He was born in Pisa, italian city famous for inclinat tower, which seems to be fall. His father was a customs officer in the town of North Africa called Bougie, so Fibonacci increased in mid-North African civilization, making, however, many trips on the Mediterranean coast. He knew so many Arab and Indian merchants and their knowledge learned arithmetic, and writing Arabic numbers. Fibonacci is known as one of the first who introduced Arabic numbers in Europe, figures that we use in our days: 0, 1, 2, 3, … 9.

A yupana is a calculator which was used by the Incas. Researchers assume that calculations were based on Fibonacci numbers to minimize the number of necessary grains per field.

Yupana_1.gif

See also:

http://plus.maths.org/issue10/features/syncopate/
http://plus.maths.org/issue29/features/quadratic/index.html
http://pagesperso-orange.fr/jean-paul.davalan/divers/fibonacci/index.html

Fibonacci series

Series of integers that, starting from third position, each term is calculated by summing the previous two, is called in mathematics of Fibonacci series:1,1,2,3,5,8,13,21,34,55,89,144…So,2=1+1; 3=1+2; 5=2+3; 8=3+5; 13=5+8; 21=13+8; 34=21+13…

About the number of gold is carefully assessing whether the series of powers of φ:φ,φ2 ,φ3…..φn,….

We know that φ2 =φ+1.Then we have :φ3 =φ2 xφ=(φ+1)φ=φ2+φ=2φ+1
φ4 =φ3 xφ=(2φ+1)φ=2φ2+φ=3φ+2

Is shown that φn=Fn x φ+Fn+1 , where Fn in the n' th term of Fibonacci's series, that any power of φ can be written with two terms in

the series of integers and of his φ =(φ1)This shows that the string of Fibonacci give coefficients developments in geomatrica's progression φ.Calculating the ratio of consecutive terms of Fibonacci series: 2/1=2; 3/2=1,5; 5/3=1,66; 8/5=1,60; 13/8=1,625; 21/13=1,615385; 43/21=1,619048; 55/34=1,617647; 89/55=1,618182; 144/89=1,617978; 233/144=1,618056.

Egyptians and pythagoras obviously used in terms of Fibonacci row and even if they had today's equipment algebra calculation, demonstrated a high knowledge and practical application of laws resonance, using integers and the relationship between them.

Divine proportion in the four kingdoms

Fibonacci's series models well many growth processes and is virtually everywhere in nature.
Linked being Sectuinea Golden Fibonacci series is the default nature of solar.It is known generally as Fibonacci's series presents a certain periodicity, considering separately the figure units, tens, hundreds etc.. And the frequency is always at the 60 (!) Number, and it was hard to follow, was also entirely Forced.
Not as forced at bringing all terms inside the decimal cycle by calculating the remainder of the division to 9 for each term.We get 24 terms, but within the cycle seems rather "stirred":1,1,2,3,5,8,4,3,7,1,8,0,8,8,7,6,4,8,5,6,2,8,1,0,…(the 24 terms).
If you believe the table but complementary terms below, in which always take place within each period by the smallest of him and "complementary "of (not that sum is 9!),production within cycle 24, two identical subcycle of 12 items:(0), 1,1,2,3,4,1,4,3,2,1,1,0 1,1,2,3,4,1,4,3,2,1,1,0 (2 periods of 12 words).

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