mgf1107lecture23 - MGF 1107 EXPLORATIONS IN MATHEMATICS...

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MGF 1107 – EXPLORATIONS IN MATHEMATICS LECTURE 23 Properties of the Fibonacci Sequence Recall from the previous lecture that the Fibonacci sequence is described by the recursive formula: F 1 = F 2 = 1, F n = F n-2 + F n-1 , n 3 It follows therefore that the first few numbers in the sequence are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, … This sequence has so many mathematical and scientific consequences that there is a journal devoted to it, Fibonacci Quarterly . We will look at some of the properties below. Property 1: The sum of the first n Fibonacci numbers is F n+2 – 1. Ex. Ex. Property 2: F 2n+1 = (F n+1 ) 2 + (F n ) 2 , n 1. Ex. Ex.
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Property 3: , n 1 Ex. Ex. Property 4: Binet’s Formula The recursive formula described earlier to crank out Fibonacci numbers is somewhat limited if we want to know the 100th number in the sequence, as it requires us to know the first 99 numbers, and it is very time consuming to calculate them all.
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This note was uploaded on 09/22/2011 for the course MAC 2311 taught by Professor Evinson during the Spring '08 term at University of Central Florida.

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mgf1107lecture23 - MGF 1107 EXPLORATIONS IN MATHEMATICS...

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