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Here, for kEZ, we will use the notation kZ = { kn|n € Z} for the set of all multiples of k. 1. The Fibonacci numbers are defined by F₁ = 1, F2 = 1, F3 = F₁+F₂ = 2, and, in general, for n ≥ 3, Fn = Fn-1 + Fn-2. Thus, the Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34,.... What is the sum and the sum F₁+F3 + + F2n+1? F2+ F₁++ F2n? Prove your answers by induction on n.

Here, for kEZ, we will use the notation kZ = { kn|n € Z} for the set of all multiples of k. 1. The Fibonacci numbers are defined by F₁ = 1, F2 = 1, F3 = F₁+F₂ = 2, and, in general, for n ≥ 3, Fn = Fn-1 + Fn-2. Thus, the Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34,.... What is the sum and the sum F₁+F3 + + F2n+1? F2+ F₁++ F2n? Prove your answers by induction on n.

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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[Number Theory] How do you solve question 1 by induction?

for the set of all multiples of k. 1. The Fibonacci numbers are defined by F₁ = 1, F2 = 1, F3 = F₁ + F₂ = 2, and, in general, for n ≥ 3, Fn = Fn-1+ Fn-2. Thus, the Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, ……. What is the sum and the sum Here, for kEZ, we will use the notation kZ = { kn|n € Z} F₁+F3 + + F2n+1? Prove your answers by induction on n. F2+ F₁+ + F2n? 2. Show that 3|n, (ii) 3 Fn 4|n, (iii) 4Fn6|n. (i) 2 Fn Hint: For example in case (ii), write Fn = 39n+rn with 0 ≤ rn <3 and consider how the Tn's are related. 3. Suppose that a and b are nonzero integers and let multiple. Show that So the sequence looks like Prove that aZn bZ = lz. 4. Suppose that a₁,..., am are integers, not all 0, and let d = gcd(a₁,..., am). Show that dZ = a₁Z+ a2Z++amZ = {a₁u₁ + +amumuj Z, 1 ≤j≤m}. 5. (i) Define a sequence of numbers Rn, n = 1,2,... Rn+1 = 5 Rn + Rn-1, lcm(a, b) be their least common by setting R₁ = 7, R₂ = 21, and for n > 2. 7,21, 112, 518, 3017,... gcd(Rn+1, Rn) = 7. (ii) Define an analogous sequence Sn with gcd(Sn+1, Sn) = 23 and justify your answer.
Transcribed Image Text:for the set of all multiples of k. 1. The Fibonacci numbers are defined by F₁ = 1, F2 = 1, F3 = F₁ + F₂ = 2, and, in general, for n ≥ 3, Fn = Fn-1+ Fn-2. Thus, the Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, ……. What is the sum and the sum Here, for kEZ, we will use the notation kZ = { kn|n € Z} F₁+F3 + + F2n+1? Prove your answers by induction on n. F2+ F₁+ + F2n? 2. Show that 3|n, (ii) 3 Fn 4|n, (iii) 4Fn6|n. (i) 2 Fn Hint: For example in case (ii), write Fn = 39n+rn with 0 ≤ rn <3 and consider how the Tn's are related. 3. Suppose that a and b are nonzero integers and let multiple. Show that So the sequence looks like Prove that aZn bZ = lz. 4. Suppose that a₁,..., am are integers, not all 0, and let d = gcd(a₁,..., am). Show that dZ = a₁Z+ a2Z++amZ = {a₁u₁ + +amumuj Z, 1 ≤j≤m}. 5. (i) Define a sequence of numbers Rn, n = 1,2,... Rn+1 = 5 Rn + Rn-1, lcm(a, b) be their least common by setting R₁ = 7, R₂ = 21, and for n > 2. 7,21, 112, 518, 3017,... gcd(Rn+1, Rn) = 7. (ii) Define an analogous sequence Sn with gcd(Sn+1, Sn) = 23 and justify your answer.
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