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.
We have a set cost of $2.25 to replace the card
So....it must be that the fines themselves must be just 6.25 - 4.25 = $4.00
So the fine per day must be just [ 4.00] / [21 - 5 ] = 4.00 / 16 = .25
a) Recursive formula
an = an-1 + .25 where a1 = .25
b) Explicit formula
an = a1 + .25(n - 1) = .25 + .25n - .25 = .25n
(c) Explicit.....it would be tedious to calculate the fine on the 60th day using the recursive formula....we would need to calculate 59 terms !!!!
(d) Fine on Day 60 = ,25(60) = $15