Recurrence Relations

1. Block 1
Establishing the Recurrence
Relation

2. What is to be learned?
• How to find the formulae for linear
recurrence relations that are not
arithmetic or geometric

3. 4 8 20 56….
Is of form Un+1 = mUn + c
so U2 = mU1 + c
U3 = mU2 + c
4m + c = 8
8m + c = 20
U1 U2 U3 U4
8 = m(4) + c
20= m(8) + c
4m = 12 m=3 c=-4
Un+1 = 3Un – 4
U5 = 3(56) – 4
Substituting
U5 = 164
Rearranging
Sim. Equations

4. Finding Recurrence Relation
2 6 26….
Is of form Un+1 = mUn + c
so U2 = mU1 + c
U3 = mU2 + c
2m + c = 6
6m + c = 26
U1 U2 U3
6 = m(2) + c
26= m(6) + c
4m = 20
m=5 c=-4
Un+1 = 5Un – 4
Substituting
Rearranging
Sim. Equations

5. 3 11 27 59….
Is of form Un+1 = mUn + c
so U2 = mU1 + c
U3 = mU2 + c
3m + c = 11
11m + c = 27
U1 U2 U3 U4
11 = m(3) + c
27 = m(11) + c
8m = 16 m=2 c=5
Un+1 = 2Un + 5
U5 = 2(59) + 5
Substituting
U5 = 123
Rearranging
Sim. Equations
Key
Question
Make a recurrence
relation to fit this
sequence and use it to
find the next term

6. An ant colony are suffering from an outbreak of
disease and worker ants are reducing by a certain
percentage weekly.
Being resourceful creatures they are capturing an
extra ants to work for them each week
There were initially 1400 worker ants.
A week later there are 1220.
The following week, there are 1076.
What percentage of ants are dying?
How many ants are they capturing?
AAn+1n+1 = mA= mAnn + c+ c
A0
A1
A2

7. AAn+1n+1 = mA= mAnn + c+ c A0 = 1400 A1 = 1220
A2 = 1076
AA11 = mA= mA00 + c+ c
AA22 = mA= mA11 + c+ c
 1220 = 1400m + c1220 = 1400m + c
 1076 = 1220m + c1076 = 1220m + c
144 = 180m
m = 0.8
c = 100
AAn+1n+1 = 0.8A= 0.8Ann + 100+ 100
20% decrease in worker ants
100 ants captured weekly

8. Sandra has borrowed money from
Lenny the loan shark.
The original loan was £3000.
She is paying back £200 each week.
She is shocked to find that 2 weeks
later she now owes £3,210!!!!!!
What are Lenny’s rates of interest?
Ln+1 = mLn + c
L0
L2
– 200
Warning - Very Difficult!!

9. Ln+1 = mLn – 200L0 = 3000
L2= 3210
Equation with L0?
L1 = mL0 – 200
L1 = m(3000) – 200
Equation with L2?
L2 = mL1 – 200
3210 = mL1 – 200
L1 = 3000m – 200
3210 = mL1 – 200

10. L1 = 3000m – 200
3210 = mL1 – 200
 3210 = m(3000m – 200) – 200
3210 = 3000m2
– 200m – 200
0 = 3000m2
– 200m – 3410
Using Big Nasty Formula
m = 1.1
Ln+1 = 1.1Ln – 200
Interest Rate 10% per week.