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(a) fn is a natural number for all natural numbers (b) fn+6= 4fn+3 + fn (c) f5n is a multiple of 5

(a) fn is a natural number for all natural numbers (b) fn+6= 4fn+3 + fn (c) f5n is a multiple of 5

Advanced Engineering Mathematics
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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3. Recall the Fibonacci sequence of numbers which is defined by f₁=1, f2 = 1, and fn+2 = fn+1+fn n ≥ 1 Use the Principle of Complete Induction to prove: (a) fn is a natural number for all natural numbers (b) fn+6 = 4fn+3 + fn (c) f5n is a multiple of 5 (d) For every natural number a fafn + fa+1fn+1 = Prove that for all n E N (e) Let o be the positive solution and p be the negative solution of the equation x² = x + 1. fn fa+n+1 on o-p (Note that is the golden rate. Solutions are = ¹+√5 and p = ¹1-√5)
Transcribed Image Text:3. Recall the Fibonacci sequence of numbers which is defined by f₁=1, f2 = 1, and fn+2 = fn+1+fn n ≥ 1 Use the Principle of Complete Induction to prove: (a) fn is a natural number for all natural numbers (b) fn+6 = 4fn+3 + fn (c) f5n is a multiple of 5 (d) For every natural number a fafn + fa+1fn+1 = Prove that for all n E N (e) Let o be the positive solution and p be the negative solution of the equation x² = x + 1. fn fa+n+1 on o-p (Note that is the golden rate. Solutions are = ¹+√5 and p = ¹1-√5)
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