**County Library**

The first day a book is overdue, you are charged 4p. Each day incurs another 4p.

What are the charges for the first week?

(4, 8, 12, 16, 20, 24, 28)

What is the Nth term?

(4N)

How much would you be charged for being 25 days late?

(100p)

**Village library**

The village library charges 10p for the first day and 3p for every subsequent day.

What are the charges for the first week?

(10, 13, 16, 19, 22, 25, 28)

What is the Nth term?

(3N+7)

What is the charge for 30 days?

(97p)

How many days late is one book if the fine is more than £2?

(Solve 3N+7>200)

Look back at both libraries. Under what conditions do the libraries have the cheapest fines?

(1-6 days: County Library

7 days: same

8+: Village library)

**Extension**

Why do the libraries have the same charge on the 7th day?

Prove it algebraically.

(Solve 3N+7=4N)

You can also extend this investigation to looking at calendar dates, with one library open 5 days a week and the other being open 6 days with fines only applying when libraries are open. How would this affect the ‘cheapness’ of fines when days are included?

**Adaptations**

This method can be used for car hire, mobile phone comparisons, energy bills because sequences link so well with graphs of real life problems.

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