TV, J4n a maitipic of 3. (b) For cach nN, f5n is a multiple of 5. (c) For each n N with n ≥ 2, f1 + ƒ2 +··· + fn−1 = fn+1 − 1. (d) For each n & N, f₁ + f3 + ... + f2n-1 f2n. (e) For each n & N, f₂ + f4+...+ f2n = f2n+1 − 1. - =
TV, J4n a maitipic of 3. (b) For cach nN, f5n is a multiple of 5. (c) For each n N with n ≥ 2, f1 + ƒ2 +··· + fn−1 = fn+1 − 1. (d) For each n & N, f₁ + f3 + ... + f2n-1 f2n. (e) For each n & N, f₂ + f4+...+ f2n = f2n+1 − 1. - =
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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Question
Transcribed Image Text:The Fibonacci sequence is defined by
F₁ = 1,
F₂ = 1, and
Fn = Fn-1+ Fn-2 for n ≥ 3.
-
The factorial numbers n! are recursively defined via 0! = 1 and n! = n · (n − 1)!
for n E N.
Transcribed Image Text:Problem 2: Section 4.3, # 2c,d,e,f
2. Assume that f₁, f2, ..., fn, ... are the Fibonacci numbers. Prove each of
the following:
★
(a) For each nN, fan is a multiple of 3.
(b) For cach n€ N, f5n is a multiple of 5.
*
* (c)
For each nN with n ≥ 2, f1 + f2 +··· + fn−1 = fn+1 − 1.
(d)
For each n€ N, f₁ + f3+ + f2n-1
f2n.
(e) For each n = N, f₂ + f4 +...+ f2n = f2n+1 − 1.
(f) For each n e N, ƒ² + ƒ2 + · ·· + ƒ2² = fnfn+1.
(g) For each nN such that n 0 (med 3), fis an odd integer.
=
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