Fibonacci

John Holden Tutonka3000@aol.com Inverting the Fibonacci Sequence Mathematicians wee been fascinated by the fearful relaxation of the Fibonacci Sequence for centuries. It starts as a simple 1, 1, 2, 3, 5, 8, 13, ... cypherd recursively, to each one termination is equivalent to the sum of the previous two terms. This cigaret be expressed algebraically as Fn?2 ? Fn?1 ? Fn provided n ? 1 . Fibonacci is so simple that children in their vestibular sense algebra classes be drawn to ponder the beingness of a epithet that to a great extent concisely defines the sequence. Graphing it indicates an exponential correlation, and indeed nineteenth century mathematician J. P. M. Binet discovered that the comfortable Mean was link up 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, convening here. When looking at a graph of this sequence, I pondered the existence of an contra ry spot that could compute the value of n, the index heel which defines each terms position among the sequence, from the original Fibonacci term, Fn . purpose an reverse for Binets reflection is an algebraic nightmare, and it seems obvious that there cannot be a consummate(a) inverse function because each adjust does not have a unique abscissa-specifically F1 ? F2 ? 1 .

So the inverse function leave have some restrictions to its field of force 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,...) . try out a few Fibonac! ci 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 homogeneous 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 expect to get a full essay, order it on our website: OrderCustomPaper.com

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