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 sumFi + F3 + ...+ F2n+1?and the sumF2 + 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 Fn3qn + rn with 0 < rn < 3 and consider how thern's are related.

Name: 1. The Fibonacci numbers are defined by F1 = 1, F2 = 1, F3 = F1+ F2 =
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Description: 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 sumFi + F3 + ...+ F2n+1?and the sumF2 + F4 + • · · + F2n

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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 %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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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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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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