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section26 - Chapter 2 MAT188H1F Lec03 Burbulla Chapter 2...

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Chapter 2 MAT188H1F Lec03 Burbulla Chapter 2 Lecture Notes Fall 2007 Chapter 2 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 2 Chapter 2 2.6: Linear Recurences Chapter 2 Lecture Notes MAT188H1F Lec03 Burbulla
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Chapter 2 2.6: Linear Recurences The Fibonacci Sequence These are a sequence of numbers, F k , defined recursively by F 1 = 1 , F 2 = 1 and F k +2 = F k +1 + F k , for k 1 . The first few terms are 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 85 , . . . The defining equation for the Fibonacci sequence is an example of a linear recurrence relation. Is there a formula for the k -th Fibonacci number? Yes; one can be found using matrices and diagonalization. I will illustrate the method for the Fibonacci sequence, but it can be applied to many other linear recurrence relations. Chapter 2 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 2 2.6: Linear Recurences A Sequence of Matrices Define the matrix V k to be V k = F k F k +1 . Then V 1 = F 1 F 2 = 1 1 and V k +1 = F k +1 F k +2 = F k +1 F k + F k +1 = 0 1 1 1 F k F k +1 = AV k , with A = 0 1 1 1 .
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