The document discusses linear recurrence relations, arithmetic sequences, and geometric sequences. It provides examples of how to write recurrence relations for various situations involving increasing or decreasing patterns over time. Examples are given of how to calculate individual terms and find a formula to represent the entire sequence. Key aspects covered include writing recurrence relations, finding subsequent terms, deriving formulas, and applying the concepts to word problems involving patterns of change over time.

1. Block 1
Linear Recurrence Relations

2. What is to be learned?
• What a linear recurrence relation is
• What an arithmetic sequence is
• What a geometric sequence is
• How to apply arithmetic and geometric
sequences

3. Linear Recurrence Relations
Ex un+1 = 2un + 3
similar to y = 2x + 3
of form un+1 = mun + c

4. Arithmetic Sequences
for un+1 = mun + c
m = 1
Ex un+1 = un + 7
un+1 = un – 6

5. Ex un+1 = un + 3 u0 = 4
Find a) U1 , U2 , U3
b) formula for Un
a) u1 = 4 + 3 = 7
u2 = 7 + 3 = 10
u3 = 13
b)
4 7 10 13
un = 3n + 4
Vital to be able to switch between the
Term no 0 1 2 3 n
3n + 4

6. Linear Recurrence Relations
of form un+1 = mun + c
Arithmetic Sequence
m = 1

7. Ex un+1 = un + 6 u0 = 2
Find a) U1 , U2 , U3
b) formula for Un
a) u1 = 2 + 6 = 8
u2 = 8 + 6 = 14
u3 = 14 + 6 = 20
b)
2 8 14 20
un = 6n + 2
Vital to switch between two types of
Term no 0 1 2 3 n
6n + 2

8. Ex Pandora is saving £3 a week for her
holidays. Kind Uncle Percy gives her
£10 to start with.
a) How much has she saved after
17 weeks?
b) How long will it take her to raise
£100?
un+1 = un +3
Or Sn+1 = Sn +3
after 1 week S1 = 13
after 2 weeks S2 = 16
u0 = 10
S0 = 10
Recurrence
Relation
Get Formula for Sn

9. Sn+1 = Sn +3 S0 = 10
Formula for Sn
10 13 16 19
Sn = 3n + 10
a) How much has she saved after
17 weeks?
n = 17
S17 = 3(17) + 10 = £61
Term no 0 1 2 3 n
3n + 10

10. Sn+1 = Sn +3 S0 = 10
Formula for Sn
10 13 16 19
Sn = 3n + 10
b) How long will it take her to raise £100?
100 = 3n + 10
90 = 3n
n = 30 weeks
Term no 0 1 2 3 n
3n + 10

11. Ex Buster does 15 sit ups daily
He decides to increase this by
2 sit ups per day
a) How many sit ups is he doing 30
days later?
b) After how many days will he be
doing 105 sit ups?
Recurrence Relation
Sn+1 = Sn + 2
S1 = 17 S2 = 19 S3 = 21
S0 = 15

12. Formula for Sn
Sequence 15 17 19 21
Formula Sn = 2n + 15
a) How many sit ups is he doing 30
days later?
n = 30
S30 = 2(30) + 15
= 75 sit ups
Term no 0 1 2 3 n
2n + 15

13. Formula for Sn
Sequence 15 17 19 21
Formula Sn = 2n + 15
b) After how many days will he be
doing 105 sit ups?
Sub Sn = 105
105 = 2n + 15
90 = 2n
n = 45 days
Term no 0 1 2 3 n
2n + 15

14. Key Question
Mr Nofair decides to build up the homework for his
class. He starts off giving them 20mins per week,
then increases this by 5 mins per week.
a) If hn is the amount of homework the class get
after n weeks, write a recurrence relation
(i.e. hn+1 = hn + ……, with h0 = …. )
b) Calculate h1 , h2 , h3 , h4
c) Find a formula for hn
d) How much homework are the class getting
after 18 weeks?
e) How many weeks will it take for the class to be
getting 4 hours of homework?

15. a) hn+1 = hn + 5, with h0 = 20
b) h1 =25 , h2 = 30 , h3 = 35, h4 = 40
c) hn = 5n + 20
d) h18 = 5(18) + 20
= 110 hours
d) 240 = 5n + 20
220 = 5n
n = 44weeks

16. What is to be learned?
• What a linear recurrence relation is
• What an arithmetic sequence is
• What a geometric sequence is
• How to apply arithmetic and geometric
sequences

17. Geometric Sequences
for un+1 = mun + c
c = 0
Ex Un+1 = 5Un

18. Percentage Increase/Decrease
Reminders
Decrease by 15%
multiply by 0.85
Increase of 15%
multiply by 1.15
 85% left
 115%

19. Ex 5p is placed on teacher’s desk
10p on desk 1, 20p on desk 2 etc.
How much will be on desk 32?
Recurrence Relation
Dn+1 = 2Dn
Formula for Dn
D0 = 5
D1 = 2 X 5
D2= 2 X 2 X 5
D0 = 5
= 23
X 5
= 22
X 5
= 21
X 5
= 20
X 5
Dn = 2n
X 5

20. Dn = 2n
X 5
Desk 32?
n = 32
D32 = 232
X 5
= £214,748,364.80

21. Geometric Sequences
for un+1 = mun + c
c = 0
Ex Un+1 = 0.8Un
Reminders
Decrease by 5%
multiply by 0.95
Increase of 5%
 95% left
 105%

22. Ex Carrie’s car is losing 30% of its value
each year.
It was worth £10 000 when she got it
How much will it be worth 5 years later?
Recurrence Relation
30% decrease  70% left
Vn+1 = 0.7 Vn V0 = 10 000

23. Formula for Vn
V0 = 10 000
V1 = 0.7 X 10 000
V2 = 0.7 X 0.7 X 10 000
V3 = 0.7 X 0.7 X 0.7 X 10 000
Vn = 0.7n
X 10 000
after 5 years
n = 5
V5 = 0.75
X 10 000
= £1 680.70
= 0.73
X 10 000
= 0.72
X 10 000
= 0.71
X 10 000
= 0.70
X 10 000