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1. On the way to the market, you exchange your cow for some magic dark chocolate espresso beans. These beans have the property that every night at midnight, each bean splits into two, effectively doubling your collection. You decide to take advantage of this and each morning (around 8am) you eat 5 beans. a. Explain why it is true that if at noon on day n you have a number of beans ending in a 5, then at noon on day n + 1 you will still have a number of beans ending in a 5. b. Why is the previous fact not enough to conclude that you will always have a number of beans ending in a 5? What additional fact would you need? c. Assuming you have the additional fact in part (b), and have successfully proved the fact in part (a), how do you know that you will always have a number of beans ending in a 5? Illustrate what is going on by carefully explaining how the two facts above prove that you will have a number of beans ending in a 5 on day 4 specifically. In other words, explain why induction works in this context.

1. On the way to the market, you exchange your cow for some magic dark chocolate espresso beans. These beans have the property that every night at midnight, each bean splits into two, effectively doubling your collection. You decide to take advantage of this and each morning (around 8am) you eat 5 beans. a. Explain why it is true that if at noon on day n you have a number of beans ending in a 5, then at noon on day n + 1 you will still have a number of beans ending in a 5. b. Why is the previous fact not enough to conclude that you will always have a number of beans ending in a 5? What additional fact would you need? c. Assuming you have the additional fact in part (b), and have successfully proved the fact in part (a), how do you know that you will always have a number of beans ending in a 5? Illustrate what is going on by carefully explaining how the two facts above prove that you will have a number of beans ending in a 5 on day 4 specifically. In other words, explain why induction works in this context.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 28EQ
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1. On the way to the market, you exchange your cow for some magic dark chocolate espresso beans. These beans have the property that every night at midnight, each bean splits into two, effectively doubling your collection. You decide to take advantage of this and each morning (around 8am) you eat 5 beans. a. Explain why it is true that if at noon on day n you have a number of beans ending in a 5, then at noon on day n + 1 you will still have a number of beans ending in a 5. b. Why is the previous fact not enough to conclude that you will always have a number of beans ending in a 5? What additional fact would you need? c. Assuming you have the additional fact in part (b), and have successfully proved the fact in part (a), how do you know that you will always have a number of beans ending in a 5? Illustrate what is going on by carefully explaining how the two facts above prove that you will have a number of beans ending in a 5 on day 4 specifically. In other words, explain why induction works in this context.
Transcribed Image Text:1. On the way to the market, you exchange your cow for some magic dark chocolate espresso beans. These beans have the property that every night at midnight, each bean splits into two, effectively doubling your collection. You decide to take advantage of this and each morning (around 8am) you eat 5 beans. a. Explain why it is true that if at noon on day n you have a number of beans ending in a 5, then at noon on day n + 1 you will still have a number of beans ending in a 5. b. Why is the previous fact not enough to conclude that you will always have a number of beans ending in a 5? What additional fact would you need? c. Assuming you have the additional fact in part (b), and have successfully proved the fact in part (a), how do you know that you will always have a number of beans ending in a 5? Illustrate what is going on by carefully explaining how the two facts above prove that you will have a number of beans ending in a 5 on day 4 specifically. In other words, explain why induction works in this context.
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