(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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![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)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd597ffd2-5c4b-4c2e-8332-77ce1607dac1%2F23603513-b781-48fc-b08e-6a7f1d20f04d%2F2vnl0e_processed.jpeg&w=3840&q=75)
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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