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.
1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,..... are part of the series. Take any number say, 89 which is the sum of two previous numbers, 34 & 55. 
Mathematically, F_n = F_n-1 + F_n-2  where F_n  is the nth term and F_n-1  is the (n-1)th term and F_n-2  is the (n-2)nd term. The series can be extended to negative numbers also, where F_-n =(-1)ⁿ⁺¹F_n

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 1-φ.

(a) φ has the property  φ= 1 + 1/ φ . In other words, it is the solution of a quadratic equation φ²  -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 C²-B²=A*D holds good.

(8) F²_n + F²_n+1 = F_2n-1  where F_n   and F_n+1    are two consecutive Fibonacci numbers and n is the  position of the number in the series.

(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/φ²) =F_n  / F_n+2

(14) x + 2x² +3x³ +5x⁴ +8x⁵+.......  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    :  Second term:  No. of terms:  Result         :   or simply fill the term number you intend to find out Find                :   th term The number is :    Formula: F(n) = [φn  -(1-φ)n]/√5	Series:     ----------- Ratio :     ----------- Variance: ----------- (from φ) No. of Entries: SD from φ:      SD % w.r.t.φ