Inverting the Fibonacci Sequence
Mathematicians wee been fascinated by the majestic 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 can be expressed algebraically as Fn?2 ? Fn?1 ? Fn provided n ? 1 . Fibonacci is so simple that children in their counterbalance algebra classes be drawn to ponder the existence of a figure that to a greater extent concisely defines the sequence. Graphing it indicates an exponential correlation, and indeed nineteenth century mathematician J. P. M. Binet discovered that the Golden Mean was related 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, rule here. When looking at a graph of this sequence, I pondered the existence of an inverse spot that could compute the value of n, the index number which defines each terms position among the sequence, from the original Fibonacci term, Fn . Finding an inverse for Binets formula is an algebraic nightmare, and it seems obvious that there cannot be a consummate(a) inverse function because each ordinate does not have a unique abscissa-specifically F1 ? F2 ? 1 .
So the inverse function leave have some restrictions to its domain 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 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 want to get a full essay, order it on our website: Orderessay
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