Higher Maths
Revision Notes
goodbye
Recurrence Relations
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Recurrence Relations
know the meaning of the terms: sequence, nth term, limit as n tends to infinity
use the notation un for
the nth term of a
sequence
interpret a recurrence
relation of the form un + 1
= mun + c
(m, c constants) in a
define a recurrence
relation of the form
un + 1 = mun + c
(m, c constants) in a
mathematical model
mathematical model
know the condition for the
limit of the sequence
resulting from a recurrence
relation to exist
find (where possible) and interpret
the limit of the sequence resulting
from a recurrence relation in a
mathematical model
A sequence is a list of terms.
The terms can be identified using: 1st, 2nd, 3rd, etc
The general term is often referred to as the nth term.
We are most interested in sequences where the nth term is a function of n.
We often use special terms for the terms of a sequence e.g.
un is often used for the nth term.
This means the 1st term is represented by u1,
The 2nd by u2 etc.
We already know how to find the formula for the nth term
where the terms increase by a constant amount.
We can list the sequence if we have a formula for un.
e.g.
e.g. 4, 11,18, 25, …
un = n2 + 2n – 1
If we assume the terms continue to increase by 7 then we
think, … the nth term is 7n … expect a 1st term of 7.
So u1 = 12 + 2.1 – 1 = 2
u2 = 22 + 2.2 – 1 = 7
u3 = 32 + 2.3 – 1 = 14
u4 = 42 + 2.4 – 1 = 23
However, the 1st term is 4
Test
Yourself?
Since 4 = 7 – 3 then the actual nth term formula is:
un = 7n – 3
When un+1 is expressed as a function of un then we have a recurrence relation.
e.g. un+1 = 3un + 4
This relation will only pin down a particular sequence if we also know one term in the
sequence, often u1, but not always.
e.g. Using the above, if u1 = 2 then u2 = 3.2 + 4 = 10; u3 = 3.10 + 4 = 34 …
giving the sequence 2, 10, 34, 106, …
Whereas , if u1 = 0 then u2 = 3.0 + 4 = 4; u3 = 3.4 + 4 = 16 …
giving the sequence 0, 4, 16, 52, …
Sometimes u0 is used which is not strictly in the sequence.
Test
Yourself?
define a recurrence relation of the form
un + 1 = mun + c
(m, c constants) in a mathematical model
(i) Given the form and some terms
choose
(ii) Given a story to model
(i) Given the form and some terms
A recurrence relation is of the form un+1 = aun + b.
If you’re given enough information, you can form a
system of two equations and solve it for a and b
Example
A recurrence relation is of the form un+1 = aun + b.
The 3rd, 4th, and 5th terms are 9, 13 and 21
respectively.
(a)
Find the recurrence relation
(b)
List the first two terms.
Using u3 = 9 and u4 = 13 and un+1 = aun + b.
13 = 9a + b … 
Using u4 = 13 and u5 = 21
21 = 13a + b …
Subtracting we  from  get 4a = 8  a = 2
Substituting in  gives 13 = 9.2 + b  b = 13 – 18 = –5
Test
Yourself?
(ii) Given a story to model
If a real-life situation is being
modelled by a recurrence
relationship take care.
The model only gives a snapshot of the actual function.
Many situations describe a
two-stage process … the model
only gives the values after both
steps have been taken.
Don’t read anthing into
apparent values at the end of
the first steps.
Example
An area initially has 5000 sites vandalised by graffiti artists.
A campaign hopes to clean 90% of the sites during the working week.
At the weekend the vandals deface another 100 sites.
Model the situation by a recurrence relation using un to represent the number
of vandalised sites on the nth Monday since the start of the campaign.
Response
un+1 = 0·1un + 100
Test
Yourself?
The condition for the limit of the sequence to exist.
What happens over time?
The recurrence relation un+1 = 0·5un + 4 with u1 = 264 produces the sequence
264, 136, 72, 40, 24, 16, 12, 10, 9, 8·5, 8·25, 8·125, …
As n tends to infinity we see that un tends towards 8.
In fact, using any starting number, this relation produces a sequence which will converge on 8.
On the other hand, the recurrence relation un+1 = 2un – 8 with u1 = 9 produces 9, 10, 12, 16, 24, 40, 72, 136, …
It diverges using any starting number … with the exception of u1 = 8, where it ‘sticks’ at 8.
In both types, 8 is called a fixed point.
•
In the first type the sequence runs towards 8 … 8 is a limit.
•
In the second type the sequence runs away from 8 … 8 is not a limit.
How do you tell the types apart?
In the relation, un+1 = aun + b, the sequence converges if –1 < a < 1 [i.e. if a is a proper fraction]
Test
Yourself?
Find and interpret the limit
Always state the grounds for a limit to exist.
If you don’t, you may just be finding a fixed point
… and every linear recurrence relation has a fixed point.
un+1 = aun + b has a limit since –1 < a < 1
How do we find the limit?
If there exists a limit, L, then, as n tends to infinity, un+1 tends to un.
Solve the equation L = aL + b for L.
How do we interpret the limit?
In any context, the recurrence relation which models it provides snapshots only of the situation.
•
Don’t make anything of ‘intermediate’ values (values deduced from the story
mid-cycle), the model doesn’t promise any sense here.
•
Don’t try fractional values of n. n is a whole number.
•
In context un itself may be a whole number to be sensible.
This will affect the interpretation of the limit.
Test
Yourself?