Recurrence Relations

1
Recurrence Relations

• 1000 is invested at an interest rate of 5 per
  annum.
• What is the value of the investment after 4
  years?
• 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.
2
Looking again at the previous problem.
The symbol we use for initial state is usually,
The initial investment was 1000.
Let us look again at the process used in (a).
Generalising this we get,
Look for the pattern and generalise to make a
formula.
3
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.
• Find a recurrence relation for the amount of drug
  in the bloodstream.
• Use your answer to calculate the amount of drug
  remaining after 24 hours.

4
(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)
5
Linear recurrence relations
6
Continuing this process,
7
Continuing this process,
8
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.
9
The mortgage problem.
A family has a mortgage of 60 000. 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.
10
The pollution problem.
The mortgage problem.
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?
11
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.
12
A limit exists because
Using the formula
Using algebra
Use whichever method you are comfortable with.
13
Using recurrence relations to find a and b.
Using simultaneous equations,
Substituting this into equation 1,