Fibonacci

John Holden Tutonka3000@aol.com Inverting the Fibonacci Sequence Mathematicians wee been fascinated by the stately relaxation of the Fibonacci Sequence for centuries. It starts as a simple 1, 1, 2, 3, 5, 8, 13, ... cypherd recursively, separately termination is equivalent to the sum of the previous two terms. This digest be expressed algebraically as Fn?2 ? Fn?1 ? Fn provided n ? 1 . Fibonacci is so simple that children in their antagonize algebra classes be drawn to ponder the human beings of a body-build that to a great extent concisely defines the sequence. Graphing it indicates an exponential correlation, and in that respectfore nineteenth century mathematician J. P. M. Binet discovered that the deluxe Mean was relate 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, hold here. When looking at a graph of this sequence, I pondered the existence of an backwa rd spot that could compute the value of n, the index fare which defines each terms position among the sequence, from the original Fibonacci term, Fn . finding an antonym for Binets decree is an algebraic nightmare, and it seems obvious that there cannot be a consummate(a) opposite function because each dedicate does not have a unique abscissa-specifically F1 ? F2 ? 1 .

So the inverse function leave have some restrictions to its vault of heaven because Binets formula does not provide a matched function. By receive I stumbled across this theoretic inverse function. It reads Fib ?1 (n) ? n ? ?log ? Fn ? ? 2 (n ? 2, 3, 4,...) . test a few Fibonacci nu! mbers 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 identical 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 compliments to get a full essay, order it on our website: OrderCustomPaper.com

If you want to get a full essay, visit our page: write my paper