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 %3D F1 + 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 + 6 | n. Hint: For example in case (ii), write Fn = 3qn + rn with 0 < rn < 3 and consider how the Tn's are related.
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 %3D F1 + 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 + 6 | n. Hint: For example in case (ii), write Fn = 3qn + rn with 0 < rn < 3 and consider how the Tn's are related.
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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![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe4829a0a-cac1-42a0-a9cc-d82a73e9ed65%2Feebf2908-1c62-4e2b-b3d6-a1fa02481eef%2F2bt8pce_processed.png&w=3840&q=75)
Transcribed Image Text: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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