Fibonacci

John Holden Tutonka3000@aol.com Inverting the Fibonacci Sequence Mathematicians wee been fascinated by the gallant relaxation of the Fibonacci Sequence for centuries. It starts as a simple 1, 1, 2, 3, 5, 8, 13, ... cypherd recursively, each termination is equivalent to the sum of the previous two terms. This squeeze out be expressed algebraically as Fn?2 ? Fn?1 ? Fn provided n ? 1 . Fibonacci is so simple that children in their discipline algebra classes be drawn to ponder the cosmos of a externalise that to a great extent concisely defines the sequence. Graphing it indicates an exponential correlation, and hence nineteenth century mathematician J. P. M. Binet discovered that the friendly Mean was link 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, get hold here. When looking at a graph of this sequence, I pondered the existence of an opposition spot that c ould compute the value of n, the index modus operandi which defines each terms position among the sequence, from the original Fibonacci term, Fn . conclusion an opponent for Binets verbalism is an algebraic nightmare, and it seems obvious that thither cannot be a consummate(a) antonym function because each ordain does not have a unique abscissa-specifically F1 ? F2 ? 1 .

So the inverse function leave have some restrictions to its orbit 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,...) . establish a few Fibonacci numb! ers 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(p) 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 regard to get a full essay, order it on our website: OrderCustomPaper.com

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