Question

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. Determine a formula for an, the nth term of this sequence. Use the formula to determine the amount of money, in dollars, Simon needs to pay when the book is 60 days late.

Answer 1

`(a)\ y = 0.25n +1`

`(b)\ \$16`

Explanation:

Given

`n \to days`

`y \to amount`

`(n_1,y_1) = (5,2.25)`

`(n_2,y_2) = (21,6.25)`

Solving (a): Formula that represents the scenario.

Calculate slope (m)

`m = (y_2 - y_1)/(n_2 -n_1)`

`m = (6.25-2.25)/(21-5)`

`m = (4)/(16)`

`m = 0.25`

The equation is calculated using:

`y = m(n - n_1) + y_1`

This gives:

`y = 0.25(n - 5) + 2.25`

`y = 0.25n - 1.25 + 2.25`

`y = 0.25n +1`

Solving (b): Amount after 60 days late.

This means that:

`n = 60`

So:

`y = 0.25 * 60 + 1`

`y = 15 + 1`

`y = 16`