3. The Fibonacci numbers, which we have discussed in class, are a sequence defined by the rules F₁ = 1, F2 = 1, and Fn = Fn-1Fn-2 for n ≥2. This problem is about the quantity Gn=Fn+2Fn+1-Fn+3Fn. (a) Compute Gn for a few values of n and make a guess about what it is in general. (b) Write down the following two expressions: • Gn+1, but with Fn+4 rewritten in terms of Fn+3 and Fn+2; Gn, but with Fn rewritten in terms of Fn+1 and Fn+2. ● 1 Now both expressions are written entirely in terms of Fn+1, Fn+2, and Fn+3, which helps see the connection between them. (c) What is the connection between the results you get, and how could you use it to prove the guess you made in part (a)?
3. The Fibonacci numbers, which we have discussed in class, are a sequence defined by the rules F₁ = 1, F2 = 1, and Fn = Fn-1Fn-2 for n ≥2. This problem is about the quantity Gn=Fn+2Fn+1-Fn+3Fn. (a) Compute Gn for a few values of n and make a guess about what it is in general. (b) Write down the following two expressions: • Gn+1, but with Fn+4 rewritten in terms of Fn+3 and Fn+2; Gn, but with Fn rewritten in terms of Fn+1 and Fn+2. ● 1 Now both expressions are written entirely in terms of Fn+1, Fn+2, and Fn+3, which helps see the connection between them. (c) What is the connection between the results you get, and how could you use it to prove the guess you made in part (a)?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Check the correction made in the problem before solving it.
Transcribed Image Text:Correction to HW problem 3
Fn-z
F₁ = F₁_₁ + F₁-₂
F₁ = 1
F₂ = 1₁
2
F3=1+1=2
F₂ =2+1=3
Fs=3+2=5
Fibonacci numbers
Transcribed Image Text:3. The Fibonacci numbers, which we have discussed in class, are a sequence defined by the
rules F₁ = 1, F2 = 1, and Fn = Fn-1Fn-2 for n ≥ 2.
This problem is about the quantity Gn = Fn+2Fn+1-Fn+3Fn.
(a) Compute Gn for a few values of n and make a guess about what it is in general.
(b) Write down the following two expressions:
• Gn+1, but with Fn+4 rewritten in terms of Fn+3 and Fn+2;
Gn, but with Fn rewritten in terms of Fn+1 and Fn+2.
1
Now both expressions are written entirely in terms of Fn+1, Fn+2, and Fn+3, which
helps see the connection between them.
(c) What is the connection between the results you get, and how could you use it to
prove the guess you made in part (a)?
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