**
Fibonacci series
A prominent torch bearer of mathematics in 13th century Europe, Leonardo
Pisano Bogollo was born in Italy in 1170 AD. The town Pisa which was his birth
place enjoys a special privilege in the realm of science. It was here at
the leaning tower of Pisa, the great Galileo nearly 250 years later,climbed to
the top and dropped two balls one light and the other very heavy and to the
utter astonishment of the assembled crowd, both hit the ground simultaneously
(the crowd expected the heavier one to hit first) thus proving a fundamental law
of falling bodies. Leonardo was nicknamed Fibonacci, meaning in Latin the "the son of Bonaccio".His
father was nicknamed 'Bonaccio' meaning a man of cheerfulness. Young
Leonardo developed an early love for mathematics. He wrote a book 'Liber
Abaci' in which, among other things, he proposed a problem on population of
breeding rabbits after a certain period, based on a set of conditions. In the
process, a number series was discovered known as Fibonacci series. This apart,
Fibonacci greatly popularized the Hindu-Arabic numeral system which ultimately
prevailed over Roman numeral system. Fibonacci died in 1250AD.
The Series : Each number (an integer) is the sum of previous two numbers, starting from 1.
1**,1,2,3,5,8,13,21,34,55,89,

Mathematically,

**Properties of the Series :**

**(1) **Take the ratio of a
Fibonacci number to its previous one(1/1,2/1,3/2,5/3,8/5 etc ). The decimal expansion of the ratio
shall progressively tend converge to a particular number called
Phi**, φ =1.61803398875 **
which is called the** Golden ratio. **
We draw a table indicating the ratios of some of the terms from the
beginning. Similarly on finding the ratio of each Fibonacci term to its
successive term (1/1,1/2,2/3,3/5,5/8 etc ),the same tends towards 0.61803398875 which is

**(a) **
**φ****
**has the property** **
**φ****=
1 + 1/ **
**φ****
. **In other words, it is the solution of a quadratic equation**
**
**φ**^{2}
-**φ****
-1 =0. **The solutions are --

** **
**φ****
= (√5 +1) / 2 ;**

** **
**φ****
= (√5 -1) / 2 ;**

**(b) Golden angle is the angle that divides 360°
or a full angle in Golden Ratio. GA=2π (1-1/φ ) =137.507°...**

**(c) **
**φ ****= 1 +Σ** _{n=1 to
n=∞}
**[-1]**^{n+1 }/ **(F**_{n}* **F**_{n+1})

**(2) **Many
modified Fibonacci series can be created with
(1,2),(1,3),(1,4)(1,5),(1,6),(1,7) as the first two members and the
series progressively tend towards phi. In fact, non-integer starting numbers
can also be used ,so called crossing sequence.

**(3) **Any series which
uses the methods of adding the last two numbers to obtain the next in the
series, will always have as a limit of the ratio of those last two numbers,
F and f. Obviously, if the numbers are squared, cubed and so forth -- i.e.
any of the numbers taken to the nth power -- we will have as limits, Fn and
fn. More importantly, if there is a separation between the numbers (where
the numbers between the ratio numbers total n), we obtain the equivalent
limits of Fn+1 and fn+1. For example, dividing 2817 by 11,933 yields 0.2361,
which in turn equals 0.618... to the 3rd (2+1) power -- the 2 representing
the separation of the two numbers, 4558 and 7375.

(**4) **The series** **numbers 1,144 are the only
squared numbers.

(**5) **The numbers 1,8 are the only cubic Fibonacci
numbers.

**(6) **The numbers 1,3,21,55 are the only triangular
Fibonacci numbers.

(**7) **For any four consecutive Fibonacci (F) numbers
A, B, C, D the relation C^{2}-B^{2}=A*D holds good.

(**8) F ^{2}**

(**9)**The square of any
F-number differs from the product of the two adjacent F numbers one on
each side by 1,sign of 1 alternating between +1 & -1.

**(10)** Every 3rd
F-number is divisible by 2, every 4th by 3,every 5th by 5,every 6th by 8.

**(11)** sequence
of the final digits repeat in cycles of 60,last two digits in cycles of 300,
final 3 digits in cycles of 1500,final 4 digits in cycles of 15000, final 5
digits in cycles of 150,000.

**(12) **Shallow** **
diagonals of Pascal's triangle sum up to give a Fibonacci number.

**(13) Phyllotactic Ratio =
(1/φ ^{2}) =F**

**(14) x + 2x ^{2} +3x^{3} +5x^{4} +8x^{5}+.......
is called Fibonacci Power Series.**

**(15) **
Starting with 5,every second Fibonacci number is
the longest side of Pythagorean triplet with other sides as integers. The
length of the longest leg of this triangle is equal to the sum of the 3
sides of the preceding triangle in this series of triangles & the shortest
leg is equal to the difference between the preceding bypassed Fibonacci
number and the shorter leg of the preceding triangle.

Generate Fibonacci Series |
Variance of ratio of any
no. to its previous no. from Phi |

First term :
or simply fill the term number you intend to find out
The number is : Formula: |
Series:
----------- Ratio : ----------- Variance: ----------- (from φ) No. of Entries: SD from φ: SD % w.r.t.φ |

**
Fibonacci Number of nth term by recursion**

**Find **
:
th term

**Result
: **
(Fibonacci no.of nth term-** F _{n}**)

**Result
: **
( nth term of the series
**Σ** _{n=1 to
n=∞}
**[-1]**^{n+1 }/ **(F**_{n}* **F**_{n+1})
)

**Series sum** :
(
**Σ** _{n=1 to
n=∞}
**[-1]**^{n+1 }/ **(F**_{n}* **F**_{n+1})
)

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