Fibonacci

John Holden Tutonka3000@aol.com Inverting the humbugonacci date Mathematicians wee been fascinated by the majestic relaxation of the Fibonacci Sequence for centuries. It starts as a plain 1, 1, 2, 3, 5, 8, 13, ... cypherd recursively, individu anyy solvent is equivalent to the sum of the forward two terms. This force out be expressed algebraically as Fn?2 ? Fn?1 ? Fn pull up stakesd n ? 1 . Fibonacci is so simple that children in their counterbalance algebra classes be drawn to ponder the human race of a figure that to a greater extent curtly defines the sequence. Graphing it indicates an exponential correlation, and indeed 19th century mathematician J. P. M. Binet discovered that the well-situated flirt with was related to the Fibonacci Sequence by proving that 1? 5 ? n ?? , provided that ? is the Golden Mean and equal in judge to . ? is Fn ? 2 ? ?? the conjugate, rule here. When facial expression at a represent of this sequence, I pondered the existence of an opposite word spot that could compute the revalue of n, the index number which defines each terms arrange among the sequence, from the original Fibonacci term, Fn . Finding an reverse for Binets formula is an algebraic nightmare, and it seems obvious that there can non be a consummate(a) inverse officiate beca expenditure each ordinate does not have a eccentric abscissa-specifically F1 ? F2 ? 1 .

So the inverse obligation leave have some restrictions to its domain because Binets formula does not provide a one-to-one serve up. By receive I stumbled crosswise this theoretic inverse function. It reads Fib ?1 (n) ? n ? ? log ? Fn ? ? 2 (n ? 2, 3, 4,...) . Try a few Fibonacci numbers yourself. use of goods and services Binets formula to uprise 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 acidify for n ? 1 because F1 ? F2 ? 1 (notice that this office the function does earn for n ? 2 ). How can we disembarrass this formula for all integers n greater than 1 ? 1? 5 . This outlet is well known...If you want to own a full essay, coif it on our website: Ordercustompaper.com

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