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3. Consider the sequence of numbers defined by the recurrence relation An = 4(an-1 – an-2), with initial terms a1 = 2 and a2 = 4. Find az through a6. What pattern do you see?

3. Consider the sequence of numbers defined by the recurrence relation An = 4(an-1 – an-2), with initial terms a1 = 2 and a2 = 4. Find az through a6. What pattern do you see?

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 57EQ
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Consider the sequence #3
1. If you have 12 boxes, then how many balls do you have to put randomly into the boxes in order to guarantee that some box will contain least 4 balls? 2. If you put 10 balls randomly into 12 boxes, is it guaranteed that no box will contain more than one ball? Why or why not? 3. Consider the sequence of numbers defined by the recurrence relation An 4(an-1 – an-2), || with initial terms a1 = 2 and a2 = 4. Find az through a6. What pattern do you see? 4. (a) Explain why 42 = 34 + 5 +3 is not a valid way of expressing 42 as a sum of non-consecutive Fibonacci numbers. (b) Modify the sum from (a) to obtain a valid way. 5. Given that the 22st Fibonacci number is 17,711, use the Golden Ratio to find the 23rd Fibonacci number. 6. Suppose a Golden Rectangle has its shorter side length equal to 5. What is the length of the longer side? What is the area of the rectangle? (You may leave your answer in terms of o.) 1
Transcribed Image Text:1. If you have 12 boxes, then how many balls do you have to put randomly into the boxes in order to guarantee that some box will contain least 4 balls? 2. If you put 10 balls randomly into 12 boxes, is it guaranteed that no box will contain more than one ball? Why or why not? 3. Consider the sequence of numbers defined by the recurrence relation An 4(an-1 – an-2), || with initial terms a1 = 2 and a2 = 4. Find az through a6. What pattern do you see? 4. (a) Explain why 42 = 34 + 5 +3 is not a valid way of expressing 42 as a sum of non-consecutive Fibonacci numbers. (b) Modify the sum from (a) to obtain a valid way. 5. Given that the 22st Fibonacci number is 17,711, use the Golden Ratio to find the 23rd Fibonacci number. 6. Suppose a Golden Rectangle has its shorter side length equal to 5. What is the length of the longer side? What is the area of the rectangle? (You may leave your answer in terms of o.) 1
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