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5. The Fibonacci sequence is defined as follows: F₁ = F2 = 1, and for each n ≥ 3, F = F-1 + F-2. Thus, the first few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34,. Prove that for any nЄ N, Fan-1 and Fan-2 are odd, while Fan is even.

5. The Fibonacci sequence is defined as follows: F₁ = F2 = 1, and for each n ≥ 3, F = F-1 + F-2. Thus, the first few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34,. Prove that for any nЄ N, Fan-1 and Fan-2 are odd, while Fan is even.

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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5. Using mathematical induction please.
of them. Prove that for any ne N, Prove that for any nЄ N, 3. Prove that for any nЄ N, 12+32 +52 + ... + (2n − 1)² = n(2n − 1)(2n+1) 1³ + 2³ + ... + n³ : ~-I(+)1' 3 4. Prove that for any nЄ N, 2 n 1 + + 1- (n + 1)! (n+1)!" 22 32 n +≤2-11 5. The Fibonacci sequence is defined as follows: F₁ = F2 = 1, and for each n ≥ 3, F = F-1 + Fn-2. Thus, the first few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, . Prove that for any nЄ N, Fan-1 and F3n-2 are odd, while F3 is even. 6. Consider a function f that is defined for all natural numbers as follows: f(1) = 1 and for each n ≥ 2, f(n) 3f(n-1)+ 10. Prove that for each natural number n, f(n) =2.3"-5. Y U B msi F10 F1 F12 PG UP DEL END PG DN 7 HOME T NEW
Transcribed Image Text:of them. Prove that for any ne N, Prove that for any nЄ N, 3. Prove that for any nЄ N, 12+32 +52 + ... + (2n − 1)² = n(2n − 1)(2n+1) 1³ + 2³ + ... + n³ : ~-I(+)1' 3 4. Prove that for any nЄ N, 2 n 1 + + 1- (n + 1)! (n+1)!" 22 32 n +≤2-11 5. The Fibonacci sequence is defined as follows: F₁ = F2 = 1, and for each n ≥ 3, F = F-1 + Fn-2. Thus, the first few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, . Prove that for any nЄ N, Fan-1 and F3n-2 are odd, while F3 is even. 6. Consider a function f that is defined for all natural numbers as follows: f(1) = 1 and for each n ≥ 2, f(n) 3f(n-1)+ 10. Prove that for each natural number n, f(n) =2.3"-5. Y U B msi F10 F1 F12 PG UP DEL END PG DN 7 HOME T NEW
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