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# 13.) Simon lost his library card and has an overdue library book.

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13.) Simon lost his library card and has an overdue library book. When the book was 5 days late, he owed $2.25 to replace his library card and pay the fine for the overdue book. When the book was 21 days late, he owed$6.25 to replace his library card and pay the fine for the overdue book. Suppose the total amount Simon owes when the book is n days late can be determined by an arithmetic sequence.

a.) Write a recursive definition to represent the amount owed when the book is n days late.

b.) Write an explicit formula, in simplest form, to represent the amount owed when the book is n days late.

c.) Simon wants to calculate how much he will owe when the book is 60 days late. Should he choose the recursive definition or the explicit formula? Explain your thinking.

d.) Calculate how much Simon owes when the book is 60 days late.

May 11, 2019
edited by GAMEMASTERX40  May 12, 2019

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$$\text{In general an arithmetic sequence has the explicit form }\\ s_n = n d + s_0\\ \text{It's recursive form is }\\ s_n = s_{n-1}+d\\ \text{In both cases }s_0 \text{ is specified}$$

$$fine_5=2.25=5d+fine_0\\ fine_{21}=6.25=21d+fine_0\\ fine_{21}-fine_5 = 4= 16d\\ d=\dfrac 1 4\\ 2.25=\dfrac 9 4 = \dfrac 5 4 + fine_0\\ fine_0=1$$

$$a) ~fine_n = fine_{n-1}+0.25,~fine_0=1\\ b)~fine_n = (0.25)n + 1$$

$$c) ~\text{Hopefully clearly he'll choose the explicit formula. Think about it.}$$

$$d)~fine_{60}=60(0.25)+1 = 16$$

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May 11, 2019