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Fibonacci Sequence

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  FIBONACCI SEQUENCE The Fibonacci sequence is a series of numbers where a number is found by adding up the two numbers before it. Starting with 0and 1, the sequence goes 0,1,1,2,3,5,8,13,21,34, ... and so forth. Written as a rule, the expression is xn =xn−1+xn−2.Named after Fibonacci, also known as Leonardo of Pisa or Leonardo Pisano, Fibonacci numbers were first introduced in his Liber abaci in 1202. The son of a Pisan merchant, Fibonacci traveled widely and traded extensively. Math was incredibly important to those in the trading industry, and his  passion for numbers was cultivated in his youth Knowledge of numbers is said to have first srcinated in the Hindu -Arabic arithmetic system, which Fibonacci studied while growing up in North Africa. Prior to the publication of Liber abaci, the Latin-speaking world had yet to be introduced to the decimal number system. He wrote many books about geometry, commercial arithmetic and irrational numbers. He also helped develop the concept of zero. The Fibonacci Sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... The next number is found by adding up the two numbers before it.    The 2 is found by adding the two numbers before it (1+1)    The 3 is found by adding the two numbers before it (1+2),    And the 5 is (2+3),    and so on! Example: the next number in the sequence above is 21+34 = 55  Here is a longer list: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, The Rule The Fibonacci Sequence can be written as a Rule First, the terms are numbered from 0 onwards like this: n =   0   1   2   3   4   5   6    7    8   9   10   11   12   13   14   ...  x n  = 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 ... So term number 6 is called x 6  (which equals 8).  Example: the 8th  term is the 7th  term plus the 6th  term: x 8  = x 7  + x 6  So we can write the rule: The Rule is x n  = x n-1  + x n-2  where:    x n  is term number n    x n-1  is the previous term (n-1)    x n-2  is the term before that (n-2) Example: term 9 is calculated like this: x 9 = x 9-1  + x 9-2  = x 8  + x 7  = 21 + 13 = 34 Leaf and flower arrangements Based on a survey of the literature encompassing 650 species and 12,500 specimens, R. Jean estimated that, among plants displaying spiral or multijugate  phyllotaxis, about 92 percent of them have Fibonacci  phyllotaxis (from the Greek: Phyllo means leaf, and taxis means arrangement). In most Aroids, a vast group of beautiful ornamental plants, flowers are arranged in a mathematical series. Clear spirals are visible and the numbers of these spirals are usually a  pair of Fibonacci numbers. For example, all the spadices of Anthurium macrolobium present floral spirals matching the Fibonacci numbers eight and five. The Indian Statistical Institute and the Royal Agri-Horticultural Society dedicated an entire study to this topic. It  provides solid evidence to supp ort these claims. Locate the lowest leaf of a green plant that hasn’t been pruned. Count both the number of times you circle the stem of the plant before arriving at the leaf located directly above the first one (pointing in the same direction), as well as the number of leaves above the lowest located leaf. The number of rotations, of turns in each direction and the number of leaves met will be Fibonacci numbers! Of course, leaf arrangements vary from species to species, but they should all be Fibonacci numbers. If the number of turns is x and the number of leaves is y, specialists commonly call the leaf arrangement x/y phyllotaxis or x/y spiral. The following ratios are the phyllotaxis ratios of different plants:   
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