Recurrence Relations £1000 is invested at an interest rate of 5% per annum. (a)What is the value of the investment after 4 years? (b)After how many years will the investment be worth £1500? A recurrence relation describes a sequence in which each term is a function of the previous terms. We can see this in (a) where the answer to year 1 is used to give us the answer to year 2 and this is used to give us year three etc.

Looking again at the previous problem. The initial investment was £1000. The symbol we use for initial state is usually, Year 1 would then beYear 2 would then be etc. Let us look again at the process used in (a). Generalising this we get, Look for the pattern and generalise to make a formula.

More complex recurrence relations We can also use recurrence relations to solve problems involving constant terms as well as variable ones. 1. A patient is injected with 160ml of a drug. Every 6 hours 25% of the drug passes out of her bloodstream. To compensate, a further 20ml dose is given every 6 hours. (a)Find a recurrence relation for the amount of drug in the bloodstream. (b)Use your answer to calculate the amount of drug remaining after 24 hours.

(b) Since we are working in 6 hour periods, 24 hours will be The amount of drug remaining in the bloodstream after 24 hours is 105ml (to the nearest ml)

Linear recurrence relations

Continuing this process,

Investigating long term effects The pollution problem. An industrial complex has requested permission to dump 50 units of chemical waste into a sea loch. It is estimated that the action of the sea will remove 40% of this waste per week. What are the long term effects of dumping this waste? The waste seems to be approaching a limit of 125 units. Hence the long term effects will be a residue of 125 units of waste in the sea loch.

The mortgage problem. A family has a mortgage of £ The interest is charged at 8% per annum. They repay £7000 each year. Examine the long term effects of the loan over time. The loan is repaid during year 16.

The pollution problem. The mortgage problem. Number of weeks Amount of waste Number of years Value of loan The graph for the pollution problem shows the sequence approaches a limit. This is said to ‘converge’ on an amount. We sometimes call this ‘tending to a limit’. The graph for the mortgage problem shows the sequence continues. This is said to ‘diverge’. It does not tend to a limit. Here we see some sequences converge whilst some diverge. Is there a rule we can use to tell if a sequence has a limit?

The Limit of a recurrence relation Proof If a limit exists then we can write As n gets very large L is the limit of the sequence.

A limit exists because Using the formula Using algebra Use whichever method you are comfortable with.

Using recurrence relations to find a and b. Using simultaneous equations, Substituting this into equation 1,