=Qkanbn!.Powers of the Q-matrix are related to the Fibonacci sequence. Observe what happens when wemultiply an arbitrary matrix byQ. We have1110! abcd!=a+cb+dab!.Multiplication of a matrix byQreplaces the first row of the matrix by the sum of the first and second25

26LECTURE 6. THE FIBONACCI Q-MATRIXrows, and the second row of the matrix by the first row.If we rewriteQitself in terms of the Fibonacci numbers asQ=F2F1F1F0!,and then make use of the Fibonacci recursion relation, we findQ2=1110! F2F1F1F0!=F3F2F2F1!.In a similar fashion,Q3is given byQ3=1110! F3F2F2F1!=F4F3F3F2!,and so on. The self-evident pattern can be seen to beQn=Fn+1FnFnFn-1!.(6.3)

27Problems for Lecture 61.Prove (6.3) by mathematical induction.2.Using the relationQnQm=Qn+m, prove the Fibonacci addition formulaFn+m=Fn-1Fm+FnFm+1.(6.4)3.Use the Fibonacci addition formula to prove the Fibonacci double angle formulasF2n-1=F2n-1+F2n,F2n=Fn(Fn-1+Fn+1).(6.5)4.Show thatF2n=LnFn.Solutions to the Problems

28LECTURE 6. THE FIBONACCI Q-MATRIX

Lecture 7Cassini’s identityView this lecture on YouTubeLast lecture’s result for the Fibonacci Q-matrix is given byQn=Fn+1FnFnFn-1!,(7.1)withQ=1110!.(7.2)From the theory of matrices and determinants (see AppendixB), we know thatdetAB=detAdetB.Repeated application of this result yieldsdetQn= (detQ)n.(7.3)Applying (7.3) to (7.1) and (7.2) results directly in Cassini’s identity (1680),Fn+1Fn-1-F2n= (-1)n.(7.4)Examples of this equality can be obtained from the first few numbers of the Fibonacci sequence1, 1, 2, 3, 5, 8, 13, 21, 34,. . . .We have2×5-32=1,3×8-52=-1,5×13-82=1,8×21-132=-113×34-212=1.Cassini’s identity is the basis of an amusing dissection fallacy, called the Fibonacci bamboozlement,discussed in the next lecture.29

30LECTURE 7. CASSINI’S IDENTITYProblems for Lecture 71.Prove Cassini’s identity by mathematical induction.2.Using the Cassini’s identity (7.4) and the Fibonacci addition formula (6.4), prove Catalan’s identityF2n-Fn-rFn+r= (-1)n-rF2r.(7.5)Solutions to the Problems

Lecture 8The Fibonacci bamboozlementView this lecture on YouTubeCassini’s identityFn+1Fn-1-F2n= (-1)ncan be interpreted geometrically:Fn+1Fn-1is the area of a rectangle of side lengthsFn+1andFn-1,andF2nis the area of a square of side lengthFn. Cassini’s identity states that the absolute difference inarea between the rectangle and the square is only one unit of area. Asnbecomes large, this one unitof area difference becomes small relative to the areas of the square and the rectangle, and Cassini’sidentity becomes the basis of an amusing dissection fallacy, called the Fibonacci bamboozlement.To perform the Fibonacci bamboozlement, one dissects a square with side lengthFnin such a waythat by rearranging the pieces, one appears able to construct a rectangle with side lengthsFn-1andFn+1, with either one unit of area larger or smaller than the original square.

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- Fall '16
- Jamie Watson
- Fibonacci number, Golden ratio