linear recurrence relations
TRANSCRIPT
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 = 1Ex 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 two types of formula
Term no 0 1 2 3 n3n + 4
Linear Recurrence Relations
of form un+1 = mun + c
Arithmetic Sequencem = 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 formula
Term no 0 1 2 3 n6n + 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 19Sn = 3n + 10
a) How much has she saved after 17 weeks?n = 17
S17 = 3(17) + 10 = £61
Term no 0 1 2 3 n3n + 10
Sn+1 = Sn +3 S0 = 10Formula for Sn
10 13 16 19Sn = 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 n3n + 10
Ex Buster does 15 sit ups dailyHe decides to increase this by 2 sit ups per daya) How many sit ups is he doing 30 days later?b) After how many days will he be doing 105 sit ups?
Recurrence RelationSn+1 = Sn + 2
S1 = 17 S2 = 19 S3 = 21
S0 = 15
Formula for Sn
Sequence 15 17 19 21Formula Sn = 2n + 15
a) How many sit ups is he doing 30 days later?
n = 30S30 = 2(30) + 15
= 75 sit ups
Term no 0 1 2 3 n2n + 15
Formula for Sn
Sequence 15 17 19 21Formula 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 n2n + 15
Key QuestionMr Nofair decides to build up the homework for hisclass. 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 hoursd) 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 = 0Ex Un+1 = 5Un
Percentage Increase/Decrease
RemindersDecrease by 15%
multiply by 0.85Increase of 15%
multiply by 1.15
85% left
115%
Ex 5p is placed on teacher’s desk10p on desk 1, 20p on desk 2 etc.How much will be on desk 32?
Recurrence RelationDn+1 = 2Dn
Formula for Dn
D0 = 5
D1 = 2 X 5
D2= 2 X 2 X 5
D3 = 2 X 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 = 0Ex Un+1 = 0.8Un
RemindersDecrease by 5%multiply by 0.95Increase of 5%multiply by 1.05
95% left
105%
Ex Carrie’s car is losing 30% of its value each year. It was worth £10 000 when she got itHow much will it be worth 5 years later?
Recurrence Relation30% decrease 70% leftVn+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 yearsn = 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