Fibonacci

John Holden Tutonka3000@aol.com Inverting the Fibonacci Sequence Mathematicians wee been fascinated by the dread relaxation of the Fibonacci Sequence for centuries. It starts as a simple 1, 1, 2, 3, 5, 8, 13, ... cypherd recursively, each(prenominal) termination is equivalent to the sum of the previous two terms. This crumb be expressed algebraically as Fn?2 ? Fn?1 ? Fn provided n ? 1 . Fibonacci is so simple that children in their equilibrate algebra classes be drawn to ponder the cosmos of a radiation pattern that to a greater extent concisely defines the sequence. Graphing it indicates an exponential correlation, and hence nineteenth century mathematician J. P. M. Binet discovered that the gilt Mean was think to the Fibonacci Sequence by proving that 1? 5 ? n ?? , provided that ? is the Golden Mean and equal in value to . ? is Fn ? 2 ? ?? the conjugate, linguistic rule here. When looking at a graph of this sequence, I pondered the existence of an opposi te word spot that could compute the value of n, the index takings which defines each terms position among the sequence, from the original Fibonacci term, Fn . purpose an opponent for Binets locution is an algebraic nightmare, and it seems obvious that in that location cannot be a consummate(a) backward function because each grade does not have a unique abscissa-specifically F1 ? F2 ? 1 .

So the inverse function leave have some restrictions to its theatre of operations because Binets formula does not provide a one-to-one function. By receive I stumbled across this theoretic inverse function. It reads Fib ? 1 (n) ? n ? ?log ? Fn ? ? 2 (n ? 2, 3, 4,...! ) . testify a few Fibonacci numbers yourself. Use Binets formula to summon the nth term, then use the new inverse function to find the index number, n , which should be the same as the first n . We already understand that the inverse function will not work for n ? 1 because F1 ? F2 ? 1 (notice that this office the function does work for n ? 2 ). How can we disembarrass this formula for all integers n greater than 1 ? 1? 5 . This result is well known...If you pauperism to get a full essay, order it on our website: OrderCustomPaper.com

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