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I have solved part A. My answer is to add numbers in front of an existing sequence that works. However, I need help with part B, which requires much higher levels of math. I know this problem has a correlation with the Fibonacci sequence(and I was thinking that I

I have solved part A. My answer is to add numbers in front of an existing sequence that works. However, I need help with part B, which requires much higher levels of math. I know this problem has a correlation with the Fibonacci sequence(and I was thinking that I

Advanced Engineering Mathematics
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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I have solved part A. My answer is to add numbers in front of an existing sequence that works. However, I need help with part B, which requires much higher levels of math. I know this problem has a correlation with the Fibonacci sequence(and I was thinking that I could maybe relate it to the closed form formula), or the correlation with golden ratio (such as the relationship between the quotient of the second term and the given term and the golden section 1.618). Please rigorously prove any answers you get, and do not a "simple" answer stating that this problem is just similar to the Fibonacci sequence and that an=an-1+an-2. 

2. Start with a positive integer, then choose a negative integer. We'll use these two numbers to generate a sequence using the following rule: create the next term in the sequence by adding the previous two. For example, if we started with 6 and -5, we would get the sequence 6,-5, 1,-4,-3, -7, -10,-17, -27,... alternating part which starts with 4 elements that alternate sign before the terms are all negative. If we started with 3 and -2, we would get the sequence 3, -2, 1,-1, 0,-1,-1, -2, -3, ... alternating part which also starts with 4 elements that alternate sign before the terms are all non-positive (we don't count 0 in the alternating part). (a) Can you find a sequence of this type that starts with 5 elements that alternate sign? With 10 elements that alternate sign? Can you find a sequence with any number of elements that alternate sign? (b) Given a particular starting integer, what negative number should you choose to make the alternating part of the sequence as long as possible? For example, if your sequence started with 8, what negative number would give the longest alternating part? What if you started with 10? With n?
Transcribed Image Text:2. Start with a positive integer, then choose a negative integer. We'll use these two numbers to generate a sequence using the following rule: create the next term in the sequence by adding the previous two. For example, if we started with 6 and -5, we would get the sequence 6,-5, 1,-4,-3, -7, -10,-17, -27,... alternating part which starts with 4 elements that alternate sign before the terms are all negative. If we started with 3 and -2, we would get the sequence 3, -2, 1,-1, 0,-1,-1, -2, -3, ... alternating part which also starts with 4 elements that alternate sign before the terms are all non-positive (we don't count 0 in the alternating part). (a) Can you find a sequence of this type that starts with 5 elements that alternate sign? With 10 elements that alternate sign? Can you find a sequence with any number of elements that alternate sign? (b) Given a particular starting integer, what negative number should you choose to make the alternating part of the sequence as long as possible? For example, if your sequence started with 8, what negative number would give the longest alternating part? What if you started with 10? With n?
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