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Block 1
Linear Recurrence Relations
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
Linear Recurrence Relations
Ex un+1 = 2un + 3
similar to y = 2x + 3
of form un+1 = mun + c
Arithmetic Sequences
for un+1 = mun + c
m = 1
Ex un+1 = un + 7
un+1 = un – 6
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
Linear Recurrence Relations
of form un+1 = mun + c
Arithmetic Sequence
m = 1
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
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
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
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
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
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
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
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?
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
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
Geometric Sequences
for un+1 = mun + c
c = 0
Ex Un+1 = 5Un
Percentage Increase/Decrease
Reminders
Decrease by 15%
multiply by 0.85
Increase of 15%
multiply by 1.15
 85% left
 115%
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
Dn = 2n
X 5
Desk 32?
n = 32
D32 = 232
X 5
= £214,748,364.80
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%
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
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

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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