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 onding in a 5? Ilustrate
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 onding in a 5? Ilustrate
Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter14: Sequences And Mathematical Induction
Section14.3: Another Look At Problem Solving
Problem 30PS
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Transcribed Image Text:a 5-cent stamp tó get make n
EXERCISES
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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