15. We define a sequence of integers as follows:fof₁fn==0,1,fn-1 + fn-2=for n ≥ 2.The integer fn is called the nth Fibonacci number. Compute the Fi-bonacci numbers fn for n = = 2,3,..., 12. Prove that (fn, fn+1) = 1for all nonnegative integers n.

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Answered: 15. We define a sequence of integers as…

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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You don't need part 1a, they are just saying it's a sequence.
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Let (an), n = 1,2,... be a positive sequence of real numbers and it satisfies a₁ + 2a2 + + nan = n(n+1)(n+2). (a) Find the values for a₁, a2, and a3. (b) Give a general expression for an (c) Prove the expression you give satisfies (1) for all n € N+ by induction.
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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 Fi + F3 + ...+ F2n+1? and the sum F2 + F4 + • · · + F2n? Prove your answers by induction on n. 2. Show that (i) 2| Fn 3 n, (ii) 3| Fn 4| n, (iii) 4| Fn A 6| n. Hint: For example in case (ii), write Fn 3qn + rn with 0 < rn < 3 and consider how the rn's are related.
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1. n+2 The sum of the first n terms of a sequence is S₁ =2-- 2" a) Determine the 1st and 2nd term of this sequence. b) Find the nth term u,. 5 3 c) Hence, find the exact value of + 32 32 + 11 2048
(4) Suppose that a, b are positive integers such that 62 – 2a? = 1. Prove that there exist positive integers c, d such that and are consecutive terms in some Farey sequence and a 1 V2 (Hint: guess expressions for c, d in terms of a, b that seem to work, then prove that they do.) b d'

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15. We define a sequence of integers as follows: fo fi fn = = 0, 1, fn-1 + fn-2 = for n ≥ 2. The integer fr is called the nth Fibonacci number. Compute the Fi- bonacci numbers fn for n = = 2,3,..., 12. Prove that (fn, fn+1) = 1 for all nonnegative integers n.