Fibonacci Numbers:1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...Define the Fibonacci sequence {f} by a recurrence relation (2nd order linear differenceequation):n=0Fibonacci matrix: F =(1) Let Xoƒn+2 = fn+1+fn, n≥0, f₁=0, f₁ =1.F-J-- [] find x₁ = Fx₁, x₂ = Fx₁, x3 = Fx₂, and Xn+1 = Fx, in terms of {f}=(3) What is F"x?(2) Find F², F³, F4 and conjecture F" in terms of {f}.Extra credit: Use Mathematical Induction to prove your formula of F".(4) Find eigenvalues of F and one eigenvector for each eigenvalue. Use the golden ratio1+√√52for your final answers.For the background of the famous number , here is one reference:https://www.britannica.com/science/golden-ratio(5) Diagonalize FOr better: Orthogonally diagonalize F: F = PDP¹, where P‐¹ = pªNote: Express P and D in terms of .(6) Find a close formula of f, using the results of the diagonalization above. Must show allwork!n+1(7) Extra credit: Using the result from (5), find lim Must show all work!n→∞(8) Find the traces of F", n=1,2,3,...fn(9) Find a recurrence relation that generates the sequence {tr (F")}~_*n=1(10) What is the famous name of the sequence{tr (F")}*__ ?n=1What kind the relation of this sequence to the Fibonacci sequence?

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Answered: Fibonacci Numbers: 1, 1, 2, 3, 5, 8,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let an be the number of strings (i.e. sequences of letters, non-sense words) of length n that can be formed using the four letters Q,X,T and Z. (a)Find an explicit formula for an (b)Find a recurrence relation that is satisfied by the numbers an. (c) Find a rational generating function (ratio of two polynomials) that is a generating function for the numbers an.
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Explain Master Theorem Using Master Theorem solve the following recurrence relation a) T(n) 3 T(n/2) + 3/4n+1.
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Consider the recurrence relation for the Fibonacci sequence and some of its initial values. FK = (-1 + Fk- 2 Fo = 1, F, = 1, F2 = 2 Use the recurrence relation and the given values for Fo, F1, and F, to compute F13 and F14: F13 F14 =
Fibonacci sequence The famous Fibonacci sequence was pro- posed by Leonardo Pisano, also known as Fibonacci, in about A.D. 1200 as a model for the growth of rabbit populations. It is given by the recurrence relation f,+1= f, + fp-1» for n = 1, 2, 3, ..., where fo = 1 and fj = 1. Each term of the sequence is the sum of its two predecessors. %3D a. Write out the first ten terms of the sequence. b. Is the sequence bounded? fa+1 c. Estimate or determine o = lim - fn ,the ratio of the successive terms of the sequence. Provide evidence that 1+ V5 , a number known as the golden mean. 2 d. Use induction to verify the remarkable result that (9" – (-1)"g"). V5
Fibonacci Numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... Define the Fibonacci sequence {f} by a recurrence relation ( 2nd order linear difference equation): Fibonacci matrix: fn+2=fn+l+fn, n≥0, fo=0, f₁=1. F-3
Find a possible function f ( n ) from the set of natural numbers to the set of natural numbers that corresponds to the sequence 7, 11, 15, 19, ... . Then find the sum of the first 20 terms of that sequence.
A sequence a,,a, ,az,..., satisfies the recurrence relation a, = 3a +10a2, with initial conditions a, = 5 and a, = 8. Find an explicit formula for the sequence. Use this formula to calculate a,0. %3D %3D
A first-order recurrence sequence is defined by the system x1 = 0, xn = 5xn-1 + 1 (n = 2, 3,4, ...) The closed form is xn = _____. (n = 1,2,3...)
A sequence is given by the recurrence relation Xn+1 = Xn + 3(n + 1) for n > 1, and the initial value x1 = = 1. (a) Showing your working, use the recurrence relation to compute n for n = 1, 2, 3 and 4. 3n2 + 3n – 4 (b) Prove for all n >1 that xn = 2
plz provide answer of alll parts (iv)(v)(vi) for part b of q3

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