PROOFS PAGE UNCORRECTED Recurrence relations · First-order recurrence relations relate a term in a...
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5 Recurrence relations 5.1 Kick off with CAS 5.2 Generating the terms of a first-order recurrence relation 5.3 First-order linear recurrence relations 5.4 Graphs of first-order recurrence relations 5.5 Review
Transcript of PROOFS PAGE UNCORRECTED Recurrence relations · First-order recurrence relations relate a term in a...
5Recurrence
relations5.1 Kick off with CAS
5.2 Generating the terms of a � rst-order recurrence relation
5.3 First-order linear recurrence relations
5.4 Graphs of � rst-order recurrence relations
5.5 Review
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ROOFS
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Please refer to the Resources tab in the Prelims section of your eBookPLUS for a comprehensive step-by-step guide on how to use your CAS technology.
5.1 Kick off with CASGenerating terms of sequences with CAS
First-order recurrence relations relate a term in a sequence to the previous term in the same sequence. You can use recurrence relations to generate all of the terms in a sequence, given a starting (or initial) value.
1 Use CAS to generate a table of the � rst 10 terms of a sequence if the starting value is 7 and each term is formed by adding 3 to the previous term.
Term Value
1 7
2
3
4
5
6
7
8
9
10
2 Use CAS to generate the � rst 10 terms of a sequence if the starting value is 14 and each term is formed by subtracting 5 from the previous term.
Term Value
1 14
2
3
4
5
6
7
8
9
10
3 Use CAS to generate the � rst 10 terms of a sequence where the starting value is 1.5 and each term is formed by multiplying the previous term by 2.
4 Use CAS to generate the � rst 10 terms of a sequence where the starting value is 4374 and each term is formed by dividing the previous term by 3.
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UNCORRECTED Use CAS to generate the � rst 10 terms of a sequence if the starting value is 14 Use CAS to generate the � rst 10 terms of a sequence if the starting value is 14
UNCORRECTED Use CAS to generate the � rst 10 terms of a sequence if the starting value is 14 and each term is formed by subtracting 5 from the previous term.
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PAGE PROOFS
PROOFSUse CAS to generate a table of the � rst 10 terms of a sequence if the starting
PROOFSUse CAS to generate a table of the � rst 10 terms of a sequence if the starting value is 7 and each term is formed by adding 3 to the previous term.
PROOFSvalue is 7 and each term is formed by adding 3 to the previous term.
Generating the terms of a first-order recurrence relationA first-order recurrence relation relates a term in a sequence to the previous term in the same sequence, which means that we only need an initial value to be able to generate all remaining terms of a sequence.
In a recurrence relation the nth term is represented by un, with the next term after un being represented by un + 1, and the term directly before un being represented by un − 1.
The initial value of the sequence is represented by either u0 or u1.
We can de� ne the sequence 1, 5, 9, 13, 17, … with the following equation:
un + 1 = un + 4 u1 = 1.
This expression is read as ‘the next term is the previous term plus 4, starting at 1’.
Or, transposing the above equation, we get:
un + 1 − un = 4 u1 = 1.
A first-order recurrence relation defines a relationship between two successive terms of a sequence, for example, between:
un, the previous term un + 1, the next term.Another notation that can be used is:
un − 1, the previous term un, the next term.The first term can be represented by either u0 or u1.
Throughout this topic we will use either notation format as short hand for next term, previous term and � rst term.
5.2
The following equations each define a sequence. Which of them are first-order recurrence relations (defining a relationship between two consecutive terms)?
a un = un − 1 + 2 u1 = 3 n = 1, 2, 3, …
b un = 4 + 2n n = 1, 2, 3, …
c fn + 1 = 3fn n = 1, 2, 3, …
THINK WRITE
a The equation contains the consecutive terms un and un − 1 to describe a pattern with a known term.
a This is a � rst-order recurrence relation. It has the pattern
un = un − 1 + 2
and a starting or � rst term of u1 = 3.
b The equation contains only the un term. There is no un + 1 or un − 1 term.
b This is not a � rst-order recurrence relation because it does not describe the relationship between two consecutive terms.
c The equation contains the consecutive terms fn and fn + 1 to describe a pattern but has no known � rst or starting term.
c This is an incomplete � rst-order recurrence relation. It has no � rst or starting term, so a sequence cannot be commenced.
WORKED EXAMPLE 111
InteractivityInitial values and � rst-order recurrence relationsint-6262
Unit 3
AOS R&FM
Topic 1
Concept 1
Generating a sequenceConcept summaryPractice questions
222 MATHS QUEST 12 FURTHER MATHEMATICS VCE Units 3 and 4
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� rst term
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recurrence relations (defining a relationship between two consecutive terms)?
u
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n −
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PAGE A first-order recurrence relation defines a relationship between
PAGE A first-order recurrence relation defines a relationship between two successive terms of a sequence, for example, between:
PAGE two successive terms of a sequence, for example, between:
PAGE u
PAGE un
PAGE n +
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Another notation that can be used is:
PAGE Another notation that can be used is:
, the previous term
PAGE , the previous term
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The first term can be represented by either PAGE The first term can be represented by either
PROOFSWe can de� ne the sequence 1, 5, 9, 13, 17, … with the following equation:
PROOFSWe can de� ne the sequence 1, 5, 9, 13, 17, … with the following equation:
This expression is read as ‘the next term is the previous term plus 4, starting at 1’.
PROOFSThis expression is read as ‘the next term is the previous term plus 4, starting at 1’.
1.
PROOFS 1.
PROOFS
PROOFS
A first-order recurrence relation defines a relationship between PROOFS
A first-order recurrence relation defines a relationship between two successive terms of a sequence, for example, between:PROOFS
two successive terms of a sequence, for example, between:
Given a fully de� ned � rst-order recurrence relation (pattern and a known term) we can generate the other terms of the sequence.
Starting termEarlier, it was stated that a starting term was required to fully de� ne a sequence. As can be seen below, the same pattern with a different starting point gives a different set of numbers.
un + 1 = un + 2 u1 = 3 gives 3, 5, 7, 9, 11, …
un + 1 = un + 2 u1 = 2 gives 2, 4, 6, 8, 10, …
Write the first five terms of the sequence defined by the first-order recurrence relation:
un = 3un − 1 + 5 u0 = 2.
THINK WRITE
1 Since we know the u0 or starting term, we can generate the next term, u1, using the pattern:
The next term is 3 × the previous term + 5.
un = 3un − 1 + 5 u0 = 2u1 = 3u0 + 5
= 3 × 2 + 5= 11
2 Now we can continue generating the next term, u2, and so on.
u2 = 3u1 + 5= 3 × 11 + 5= 38
u3 = 3u2 + 5= 3 × 38 + 5= 119
u4 = 3u3 + 5= 3 × 119 + 5= 362
3 Write your answer. The sequence is 2, 11, 38, 119, 362.
WORKED EXAMPLE 222
A sequence is defined by the first-order recurrence relation:
un + 1 = 2un − 3 n = 1, 2, 3, …
If the fourth term of the sequence is −29, that is, u4 = −29, then what is the second term?
THINK WRITE
1 Transpose the equation to make the previous term, un, the subject.
un =un + 1 + 3
2
2 Use u4 to � nd u3 by substituting into the transposed equation.
u3 = u4 + 32
= −29 + 32
= −13
WORKED EXAMPLE 333
Topic 5 RECURRENCE RELATIONS 223
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A sequence is defined by the first-order recurrence relation:
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PAGE = 11
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u
PAGE u2
PAGE 2 =
PAGE = 3
PAGE 3u
PAGE u
=
PAGE = 3
PAGE 3
PROOFS
PROOFSWrite the first five terms of the sequence defined by the first-order
PROOFSWrite the first five terms of the sequence defined by the first-order
PROOFS
PROOFS
1 PROOFS
1 + PROOFS
+ 5 PROOFS
5 uPROOFS
u5 PROOFS
55PROOFS
5PROOFS
Generating the terms of a first-order recurrence relation1 WE1 The following equations each de�ne a sequence. Which of them
are �rst-order recurrence relations (de�ning a relationship between two consecutive terms)?
a un = un − 1 + 6 u1 = 7 n = 1, 2, 3, ...
b un = 5n + 1 n = 1, 2, 3, ...
2 The following equations each de�ne a sequence. Which of them are �rst-order recurrence relations?
a un = 4un − 1 − 3 n = 1, 2, 3, ...
b fn + 1 = 5fn − 8 f1 = 0 n = 1, 2, 3, ...
3 WE2 Write the �rst �ve terms of the sequence de�ned by the �rst-order recurrence relation:
un = 4un − 1 + 3 u0 = 5.
4 Write the �rst �ve terms of the sequence de�ned by the �rst-order recurrence relation:
fn + 1 = 5fn − 6 f0 = −2.
5 WE3 A sequence is de�ned by the �rst-order recurrence relation:
un + 1 = 2un − 1 n = 1, 2, 3, ...
If the fourth term of the sequence is 5, that is, u4 = 5, then what is the second term?
6 A sequence is de�ned by the �rst-order recurrence relation:
un + 1 = 3un + 7 n = 1, 2, 3, ...
If the seventh term of the sequence is 34, that is, u7 = 34, then what is the �fth term?
7 Which of the following equations are complete �rst-order recurrence relation?
a un = 2 + nb un = un − 1 − 1 u0 = 2
c un = 1 − 3un − 1 u0 = 2d un − 4un − 1 = 5
e un = −un − 1f un = n + 1 u1 = 2
g un = 1 − un − 1 u0 = 21h un = an − 1 u2 = 2
i fn + 1 = 3fn − 1j pn = pn − 1 + 7 u0 = 7
EXERCISE 5.2
PRACTISE
CONSOLIDATE
3 Use u3 to �nd u2. u2 = u3 + 32
= −13 + 32
= −5
4 Write your answer. The second term, u2, is −5.
224 MATHS QUEST 12 FURTHER MATHEMATICS VCE Units 3 and 4
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f
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If the fourth term of the sequence is 5, that is,
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A sequence is de�ned by the �rst-order recurrence
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If the seventh term of the sequence is 34, that is,
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�fth term?
7
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PAGE Write the �rst �ve terms of the sequence de�ned by the �rst-order
PAGE Write the �rst �ve terms of the sequence de�ned by the �rst-order
n
PAGE n−
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Write the �rst �ve terms of the sequence de�ned by the �rst-order
PROOFSrelation
PROOFSrelation
relations (de�ning a relationship between two
PROOFSrelations (de�ning a relationship between two
The following equations each de�ne a sequence. Which of them are �rst-order
PROOFSThe following equations each de�ne a sequence. Which of them are �rst-order
8 Write the �rst �ve terms of each of the following sequences.a un = un − 1 + 2 u0 = 6 b un = un − 1 − 3 u0 = 5
c un = 1 + un − 1 u0 = 23 d un + 1 = un − 10 u1 = 7
9 Write the �rst �ve terms of each of the following sequences.a un = 3un − 1 u0 = 1 b un = 5un − 1 u0 = −2c un = −4un − 1 u0 = 1 d un + 1 = 2un u1 = −1
10 Write the �rst �ve terms of each of the following sequences.
a un = 2un − 1 + 1 u0 = 1 b un = 3un − 1 − 2 u1 = 5
11 Write the �rst seven terms of each of the following sequences.
a un = −un − 1 + 1 u0 = 6 b un + 1 = 5un − 1 u1 = 1
12 Which of the sequences is generated by the following �rst-order recurrence relation?
un = 3un − 1 + 4 u0 = 2
A 2, 3, 4, 5, 6, … B 2, 6, 10, 14, 18, …
C 2, 10, 34, 106, 322, … D 2, 11, 47, 191, 767, …
E 6, 10, 14, 18, 22, …
13 Which of the sequences is generated by the following �rst-order recurrence relation?
un + 1 = 2un − 1 u1 = −3
A −3, 5, 9, 17, 33, … B −3, −5, −9, −17, −33, …
C −3, 5, −3, 5, −3, … D −3, −8, −14, −26, −54, …
E −3, −7, −15, −31, −63, …
14 A sequence is de�ned by the �rst-order recurrence relation:
un + 1 = 3un + 1 n = 1, 2, 3, …
If the fourth term is 67 (that is, u4 = 67), what is the second term?
15 A sequence is de�ned by the �rst-order recurrence relation:
un + 1 = 4un − 5 n = 1, 2, 3, …
If the third term is −41 (that is, u3 = −41), what is the �rst term?
16 For the sequence de�ned in question 15, if the seventh term is −27, what is the �fth term?
17 A sequence is de�ned by the �rst-order recurrence relation:
un + 1 = 5un − 10 n = 1, 2, 3, …
If the third term is −10, the �rst term is:
A −146
B 56
C 0 D 2 E 4
18 Write the �rst-order recurrence relations for the following descriptions of a sequence and generate the �rst �ve terms of the sequence.
a The next term is 3 times the previous term, starting at 14
.
b Next year’s attendance at a motor show is 2000 more than the previous year’s attendance, with a �rst year attendance of 200 000.
c The next term is the previous term less 7, starting at 100.d The next day’s total sum is double the previous day’s sum less 50, with a �rst
day sum of $200.
MASTER
Topic 5 RECURRENCE RELATIONS 225
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PAGE Which of the sequences is generated by the following �rst-order recurrence
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PAGE un
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A sequence is de�ned by the �rst-order recurrence
PROOFS
PROOFS
PROOFS u
PROOFSu1
PROOFS1 =
PROOFS= 1
PROOFS 1
Which of the sequences is generated by the following �rst-order recurrence
PROOFSWhich of the sequences is generated by the following �rst-order recurrence
2
PROOFS 2
2, 6, 10, 14, 18, …
PROOFS2, 6, 10, 14, 18, …2, 11, 47, 191, 767, …
PROOFS2, 11, 47, 191, 767, …
Which of the sequences is generated by the following �rst-order recurrencePROOFS
Which of the sequences is generated by the following �rst-order recurrence
First-order linear recurrence relationsFirst-order linear recurrence relations with a common differenceConsider the sequence 3, 7, 11, 15, 19, …
The common difference, d, is the value between consecutive terms in the sequence: d = u2 − u1 = u3 − u2 = u4 − u3 = …
d = 7 − 3 = 11 − 7 = 15 − 11 = +4
The common difference is +4.
This sequence may be de� ned by the � rst-order linear recurrence relation:
un + 1 − un = 4 u1 = 3.
Rewriting this, we obtain:
un + 1 = un + 4 u1 = 3.
A sequence with a common difference of d may be defined by a first-order linear recurrence relation of the form:
un + 1 = un + d (or un + 1 − un = d)
where d is the common difference and for
d > 0 it is an increasing sequence
d < 0 it is a decreasing sequence.
5.3
Express each of the following sequences as first-order recurrence relations.
a 7, 12, 17, 22, 27, …
b 9, 3, −3, −9, −15, …
THINK WRITE
a 1 Write the sequence. a 7, 12, 17, 22, 27, …
2 Check for a common difference. d = u4 − u3
= 22 − 17= 5
d = u3 − u2
= 17 − 12= 5
d = u2 − u1
= 12 − 7= 5
3 There is a common difference of 5 and the � rst term is 7.
The � rst-order recurrence relation is given by:un+1 = un + dun+1 = un + 5 u1 = 7
b 1 Write the sequence. b 9, 3, −3, −9, −15, …
2 Check for a common difference. d = u4 − u3
= −9 − −3= −6
d = u3 − u2
= −3 − 3= −6
d = u2 − u1
= 3 − 9= −6
3 There is a common difference of −6 and the � rst term is 9.
The � rst-order recurrence relation is given by:un + 1 = un − 6 u1 = 9
WORKED EXAMPLE 444
InteractivityFirst-order recurrence relations with a common differenceint-6264
Unit 3
AOS R&FM
Topic 1
Concept 2
First-order linear recurrence relationsConcept summaryPractice questions
226 MATHS QUEST 12 FURTHER MATHEMATICS VCE Units 3 and 4
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7, 12, 17, 22, 27, …
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(or
PAGE (or u
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PAGE 1
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0 it is a decreasing sequence.PAGE 0 it is a decreasing sequence.
PROOFSrelation:
PROOFSrelation:
PROOFS
PROOFS
d PROOFS
d may be defined by PROOFS
may be defined by d may be defined by d PROOFS
d may be defined by drelation of the form:PROOFS
relation of the form:
Express the following sequence as a first-order recurrence relation.
un = −3n − 2 n = 1, 2, 3, 4, 5, …
THINK WRITE
1 Generate the sequence using the given rule. n = 1, 2, 3, 4, 5, … un = −3n − 2n = 1 u1 = −3 × 1 − 2
= −3 − 2= −5
n = 2 u2 = −3 × 2 − 2= −6 − 2= −8
n = 3 u3 = −3 × 3 − 2= −9 − 2= −11
n = 4 u4 = −3 × 4 − 2= −12 − 2= −14
The sequence is −5, −8, −11, −14, …
2 There is a common difference of −3 and the � rst term is −5.
Write the � rst-order recurrence relation.
The � rst order difference equation is:un + 1 = un − 3 u1 = −5
WORKED EXAMPLE 555
First-order linear recurrence relations with a common ratioNot all sequences have a common difference (i.e. a sequence increasing or decreasing by adding or subtracting the same number to � nd your next term). You might also � nd that a sequence increases or decreases by multiplying the terms by a common ratio.
Consider the geometric 1, 3, 9, 27, 81, … The common ratio is found by:
R =u2
u1=
u3
u2=
u4
u3= . . .
R = 31
= 93
= 279
= . . .
= 3The common ratio is 3.
This sequence may be de� ned by the � rst-order linear recurrence relation:
un + 1 = 3un u1 = 1
A sequence with a common ratio of R may be defined by a first-order linear recurrence relation of the form:
un + 1 = Run
where R is the common ratioR > 1 is an increasing sequence0 < R < 1 is a decreasing sequence R < 0 is a sequence alternating between positive and negative values.
InteractivityFirst-order recurrence relations with a common ratioint-6263
Topic 5 RECURRENCE RELATIONS 227
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UNCORRECTED Not all sequences have a common difference (i.e. a sequence increasing or decreasing by adding or subtracting the same number to � nd your next term). You might also � nd
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PAGE
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PAGE The sequence is
The � rst order difference equation is:
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1
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PAGE un
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First-order linear recurrence relations with a common ratioPAGE
First-order linear recurrence relations with a common ratio
PROOFS2
PROOFS2
6
PROOFS6 −
PROOFS− 2
PROOFS2= −
PROOFS= −8
PROOFS8
3
PROOFS3 = −
PROOFS= −3
PROOFS3 ×
PROOFS× 3
PROOFS3
= −
PROOFS= −9
PROOFS9
= −
PROOFS= −
4
PROOFS 4 u
PROOFSu4
PROOFS4
5, PROOFS
5,
Express each of the following sequences as first-order recurrence relations.
a 1, 5, 25, 125, 625, … b 3, −6, 12, −24, 48 …
THINK WRITE
a 1 There is a common ratio of 5 and the � rst term is 1.
a R = 51
= 255
= 12525
= …
= 5
u1 = 1
2 Write the � rst-order recurrence relation. The � rst-order recurrence relation is given by:un + 1 = 5un u1 = 1
b 1 There is a common ratio of −2 and the � rst term is 3.
b R = −63
= 12−6
= −2412
= …
= −2
u1 = 3
2 Write the � rst-order recurrence relation. The � rst-order recurrence relation is given by:un + 1 = −2un u1 = 3
WORKED EXAMPLE 666
Express each of the following sequences as first-order recurrence relations.
a un = 2(7)n − 1 n = 1, 2, 3, 4, …
b un = −3(2)n − 1 n = 1, 2, 3, 4, …
THINK WRITE
a 1 Generate the sequence using the given rule. a n = 1, 2, 3, 4, … un = 2(7)n − 1
n = 1 u1 = 2(7)1 − 1
= 2 × 70
= 2 × 1
= 2n = 2 u1 = 2(7)2 − 1
= 2 × 71
= 2 × 7
= 14n = 3 u1 = 2(7)3 − 1
= 2 × 72
= 2 × 49
= 98n = 4 u1 = 2(7)4 − 1
= 2 × 73
= 2 × 343
= 686
2 There is a common ratio of 7 and the � rst term is 2.
R = 7, u1 = 2
WORKED EXAMPLE 777
228 MATHS QUEST 12 FURTHER MATHEMATICS VCE Units 3 and 4
c05RecurrenceRelations.indd 228 12/09/15 4:08 AM
UNCORRECTED 1, 2, 3, 4, …
UNCORRECTED 1, 2, 3, 4, …
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED WRITE
UNCORRECTED WRITE
Generate the sequence using the given rule.
UNCORRECTED Generate the sequence using the given rule.
PAGE
PAGE
PAGE n
PAGE n
PAGE u
PAGE u1
PAGE 1
PAGE Express each of the following sequences as first-order recurrence relations.
PAGE Express each of the following sequences as first-order recurrence relations.
1, 2, 3, 4, …
PAGE 1, 2, 3, 4, …
1, 2, 3, 4, …PAGE
1, 2, 3, 4, …PAGE PROOFSrelation is given by:
PROOFSrelation is given by:
The � rst-order recurrencePROOFS
The � rst-order recurrence=PROOFS
= 3PROOFS
3
3 Write the �rst-order recurrence relation. The �rst-order recurrence relation is given by:un + 1 = 7un u1 = 2
b 1 Generate the sequence using the given rule. b n = 1, 2, 3, 4, … un = −3(2)n − 1
n = 1 u1 = −3(2)1 − 1
= −3 × 20
= −3 × 1
= −3n = 2 u1 = −3(2)2 − 1
= −3 × 21
= −3 × 2
= −6n = 3 u1 = −3(2)3 − 1
= −3 × 22
= −3 × 4
= −12n = 4 u1 = −3(2)4 − 1
= −3 × 23
= −3 × 8
= −24
2 There is a common ratio of 2 and the �rst term is −3.
R = 2, u1 = −3
3 Write the �rst-order recurrence relation. The �rst-order recurrence relation is given by:un + 1 = 2un u1 = −3
First-order linear recurrence relations1 WE4 Express each of the following sequences as �rst-order recurrence relations.
a 1, 4, 7, 10, 13, … b 12, 8, 4, 0, −4, …
2 Express each of the following sequences as �rst-order recurrence relations.
a −24, −19, −14, −9, −4, … b 55, 66, 77, 88, 99, …
3 WE5 Express the following sequence as a �rst-order recurrence relation.
un = −5n − 3 n = 1, 2, 3, 4, 5, …
4 Express the sequence as a �rst-order recurrence relation.
un = 12
n + 4 n = 1, 2, 3, 4, 5, …
5 WE6 Express each of the following sequences as �rst-order recurrence relations.
a 2, 10, 20, 40, 80, … b 4, −8, 16, −32, 64, …
6 Express each of the following sequences as �rst-order recurrence relations.
a 3, 15, 45, 225, 1125, … b 200, −100, 50, −25, 12.5, …
7 WE7 Express each of the following sequences as �rst-order recurrence relations.
a un = 2(5)n − 1 n = 1, 2, 3, 4, … b un = −3(4)n − 1 n = 1, 2, 3, 4, …
EXERCISE 5.3
PRACTISE
Topic 5 RECURRENCE RELATIONS 229
c05RecurrenceRelations.indd 229 12/09/15 4:08 AM
UNCORRECTED
UNCORRECTED
UNCORRECTED There is a common ratio of 2 and the �rst
UNCORRECTED There is a common ratio of 2 and the �rst
Write the �rst-order recurrence relation.
UNCORRECTED Write the �rst-order recurrence relation.
First-order linear recurrence
UNCORRECTED
First-order linear recurrence
UNCORRECTED
WE4
UNCORRECTED
WE4 Express each of the following sequences as �rst-order recurrence relations.
UNCORRECTED
Express each of the following sequences as �rst-order recurrence relations.
a
UNCORRECTED
a 1, 4, 7, 10, 13, …
UNCORRECTED
1, 4, 7, 10, 13, …
2
UNCORRECTED
2 Express each of the following sequences as �rst-order recurrence relations.
UNCORRECTED
Express each of the following sequences as �rst-order recurrence relations.
a
UNCORRECTED
a
PAGE
PAGE
R PAGE
R
PROOFS2
PROOFS21
PROOFS1
3
PROOFS3 ×
PROOFS× 2
PROOFS2
= −
PROOFS= −6
PROOFS6
u
PROOFSu1
PROOFS1 = −
PROOFS= −3(2)
PROOFS3(2)3
PROOFS3
= −
PROOFS= −3
PROOFS3
8 Express each of the following sequences as �rst-order recurrence relations.
a un = 3(2)n − 1 n = 1, 2, 3, 4, … b un = −2(3)n − 1 n = 1, 2, 3, 4, …
9 Express each of the following sequences as �rst-order recurrence relations.
a 1, 3, 5, 7, 9, … b 3, 10, 17, 24, 31, … c 12, 5, −2, −9, −16, …d 1, 0.5, 0, −0.5, −1, … e 2, 6, 10, 14, 18, … f −2, 2, 6, 10, 14, …g 6, 1, −4, −9, −14, … h 4, 10.5, 17, 23.5, 30, …
10 The sequence −6, −3, 0, 3, 6, … can be de�ned by the �rst-order recurrence relation:
A un + 1 = un − 3 u0 = −6 B un + 1 = un + 3 u1 = −6C un + 1 = 3un u1 = −3 D un + 1 = 3un − 1 u0 = −3E un + 1 = 3un u1 = 3
11 Express each of the following sequences as �rst-order recurrence relations.
a un = −n − 3 n = 1, 2, 3, … b un = 2n + 1 n = 1, 2, 3, …
c un = 3n − 4 n = 1, 2, 3, … d un = −2n + 6 n = 1, 2, 3, …
12 The sequence de�ned by un = −2n + 3, n = 1, 2, 3, …, can be de�ned by the �rst-order recurrence relation:
A un + 1 = un − 2 u1 = 1 B un + 1 = −2un u1 = 3C un + 1 = −2un un = 1 D un + 1 = un + 3 u1 = 1E un + 1 = −2un + 3 u1 = 1
13 Express each of the following sequences as a �rst-order recurrence relation.
a 12, 10, 8, 6, 4, … b 1, 2, 3, 4, 5, 6, …
14 Express each of the following sequences as �rst-order recurrence relations.
a 5, 10, 20, 40, 80, … b 1, 6, 36, 216, 1296, …c −3, 3, −3, 3, −3, … d −3, −12, −48, −192, −768, …e 2, 6, 18, 54, 162, … f 5, −5, 5, −5, 5, …g −2, −8, −32, −128, −512, … h 5, −15, 45, −135, 405, …
15 The sequence −2, 6, −18, 54, −162, … can be de�ned by the �rst-order recurrence relation:
A un + 1 = −2un u1 = −2 B un + 1 = −2un u1 = 3C un + 1 = −3un u1 = 3 D un + 1 = −3un u1 = −2E un + 1 = 3un u1 = 2
16 Express each of the following sequences as �rst-order recurrence relations.
a un = 2(3)n − 1 n = 1, 2, 3, … b un = −3(4)n − 1 n = 1, 2, 3, …
c un = 0.5(−1)n − 1 n = 1, 2, 3, … d un = 3(5)n − 1 n = 1, 2, 3, …e un = −5(2)n − 1 n = 1, 2, 3, … f un = 0.1(−3)n − 1 n = 1, 2, 3, …
17 The sequence un = −4(1)n − 1 n = 1, 2, 3, … can be de�ned by the �rst-order recurrence relation:
A un + 1 = un u1 = −4 B un + 1 = un − 1 u1 = −4C un + 1 = 4un u1 = 1 D un + 1 = −4un + 3 u1 = −4E un + 1 = −4un u1 = −1
18 Express each of the following sequences as a �rst-order recurrence relation.
a −4, 4, −4, 4, −4, … b 2, −14, 98, −686, 4802, …
c 2, 6, 18, 54, 162, … d −1, 1, −1, 1, −1, …
e 5, 10, 20, 40, 80, …
CONSOLIDATE
230 MATHS QUEST 12 FURTHER MATHEMATICS VCE Units 3 and 4
c05RecurrenceRelations.indd 230 12/09/15 4:08 AM
UNCORRECTED Express each of the following sequences as �rst-order recurrence relations.
UNCORRECTED Express each of the following sequences as �rst-order recurrence relations.
2, 6, 18, 54, 162, …
UNCORRECTED 2, 6, 18, 54, 162, …
−
UNCORRECTED −128,
UNCORRECTED 128, −
UNCORRECTED −512, …
UNCORRECTED 512, …
The sequence
UNCORRECTED The sequence −
UNCORRECTED −2, 6,
UNCORRECTED 2, 6, −
UNCORRECTED −18, 54,
UNCORRECTED 18, 54,
recurrence relation:
UNCORRECTED
recurrence relation:
1
UNCORRECTED
1 =
UNCORRECTED
= −
UNCORRECTED
−2
UNCORRECTED
2u
UNCORRECTED
un
UNCORRECTED
n
n
UNCORRECTED
n +
UNCORRECTED
+ 1
UNCORRECTED
1 =
UNCORRECTED
= −
UNCORRECTED
−3
UNCORRECTED
3u
UNCORRECTED
uE
UNCORRECTED
E u
UNCORRECTED
un
UNCORRECTED
n +
UNCORRECTED
+ 1
UNCORRECTED
1 =
UNCORRECTED
= 3
UNCORRECTED
3
16
UNCORRECTED
16 Express each of the following sequences as �rst-order recurrence relations.
UNCORRECTED
Express each of the following sequences as �rst-order recurrence relations.
a
UNCORRECTED
a u
UNCORRECTED
uc
UNCORRECTED
c
PAGE u
PAGE un
PAGE n +
PAGE + 1
PAGE 1
D
PAGE D u
PAGE un
PAGE n +
PAGE + 1
PAGE 1 =
PAGE =Express each of the following sequences as a �rst-order recurrence relation.
PAGE Express each of the following sequences as a �rst-order recurrence relation.
Express each of the following sequences as �rst-order recurrence relations.PAGE
Express each of the following sequences as �rst-order recurrence relations.
PROOFS=
PROOFS= −
PROOFS−
u
PROOFSu0
PROOFS0 =
PROOFS= −
PROOFS−3
PROOFS3Express each of the following sequences as �rst-order recurrence relations.
PROOFSExpress each of the following sequences as �rst-order recurrence relations.
PROOFS n
PROOFSn =
PROOFS= 1, 2, 3, …
PROOFS 1, 2, 3, …
+
PROOFS+ 6
PROOFS 6
PROOFS n
PROOFSn
1, 2, 3, …, can be de�ned by the
PROOFS 1, 2, 3, …, can be de�ned by the
−PROOFS
−2PROOFS
2
19 Express each of the following sequences as recurrence relations.
a 4, 10.5, 17, 23.5, 30, … b 15, 10, 5, 0, −5, −10, …c 1, 5, 9, 13, 17, 21, …
20 Express each of the geometric sequences as recurrence relations.
a un = 4(2)n – 1, n = 1, 2, 3, … b un = −3(4)n – 1, n = 1, 2, 3, …c un = 2(−6)n – 1, n = 1, 2, 3, … d un = −0.1(8)n – 1, n = 1, 2, 3, …e un = 3.5(−10)n – 1, n = 1, 2, 3, …
Graphs of first-order recurrence relationsCertain quantities in nature and business may change in a uniform way (forming a pattern).
This change may be an increase, as in the case of:un + 1 = un + 2 u1 = 3,
or it may be a decrease, as in the case of:un + 1 = un − 2 u1 = 3.
These patterns can be modelled by graphs that, in turn, can be used to recognise patterns in the real world.
A graph of the equation could be drawn to represent a situation, and by using the graph the situation can be analysed to � nd, for example, the next term in the pattern.
First-order recurrence relations: un + 1 = un + b (arithmetic patterns)The sequences of a � rst-order recurrence relation un + 1 = un + b are distinguished by a straight line or a constant increase or decrease.
un
n20
1
41 3 5
2345
Val
ue o
f te
rm
Term number
un
n20
1
41 3 5
2345
Val
ue o
f te
rm
Term number
MASTER
5.4
An increasing pattern or a positive common difference gives an upward straight line.
A decreasing pattern or a negative common difference gives a downward straight line.
On a graph, show the first five terms of the sequence described by the first-order recurrence relation:
un + 1 = un − 3 u1 = −5.
THINK WRITE/DRAW
1 Generate the values of each of the � ve terms of the sequence.
un + 1 = un − 3 u1 = −5u2 = u1 − 3
= −5 − 3= −8
u3 = u2 − 3
= −8 − 3= −11
u4 = u3 − 3= −11 − 3= −14
u5 = u4 − 3
= −14 − 3= −17
WORKED EXAMPLE
Topic 5 RECURRENCE RELATIONS 231
c05RecurrenceRelations.indd 231 12/09/15 4:08 AM
UNCORRECTED The sequences of a � rst-order recurrence
UNCORRECTED The sequences of a � rst-order recurrencea straight line or a constant increase or decrease.
UNCORRECTED a straight line or a constant increase or decrease.
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
0
UNCORRECTED
0
1
UNCORRECTED
1
UNCORRECTED
1 3 5
UNCORRECTED
1 3 5
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
An increasing pattern or a positive
UNCORRECTED
An increasing pattern or a positive common difference gives an upward
UNCORRECTED
common difference gives an upward straight line.
UNCORRECTED
straight line.
UNCORRECTED
UNCORRECTED
WORKED
UNCORRECTED
WORKEDEXAMPLE
UNCORRECTED
EXAMPLE
UNCORRECTED
UNCORRECTED
UNCORRECTED PAGE can be used to recognise patterns in the real world.
PAGE can be used to recognise patterns in the real world.
A graph of the equation could be drawn to represent a situation, and by using the
PAGE A graph of the equation could be drawn to represent a situation, and by using the graph the situation can be analysed to � nd, for example, the next term in the pattern.
PAGE graph the situation can be analysed to � nd, for example, the next term in the pattern.
First-order recurrence relations:
PAGE First-order recurrence relations:
The sequences of a � rst-order recurrencePAGE
The sequences of a � rst-order recurrencea straight line or a constant increase or decrease.PAGE
a straight line or a constant increase or decrease.
PROOFSGraphs of first-order recurrence relations
PROOFSGraphs of first-order recurrence relations
These patterns can be modelled by graphs that, in turn, PROOFS
These patterns can be modelled by graphs that, in turn, can be used to recognise patterns in the real world. PROOFS
can be used to recognise patterns in the real world. PROOFS
First-order recurrence relations: un + 1 = RunThe sequences of a � rst-order recurrence relation un + 1 = Run are distinguished by a curved line or a saw form.
un
n20
1
41 3 5 6
2345
Val
ue o
f te
rm
Term number
6
n
un
20
1
4 61 3 5
23456
Val
ue o
f te
rm
Term number
An increasing pattern or a positive common ratio greater than 1 (R > 1) gives an upward curved line.
A decreasing pattern or a positive fractional common ratio (0 < R < 1) gives a downward curved line.
un
n0 41 3 5 6
−8−6−4−2
Val
ue o
f te
rm
Term number
2468
10
un
n0 2 41 3 5 6
−8−6−4−2
Val
ue o
f te
rm
Term number
2468
10
An increasing saw pattern occurs when the common ratio is a negative value less than −1 (R < −1).
A decreasing saw pattern occurs when the common ratio is a negative fraction (−1 < R < 0).
On a graph, show the first six terms of the sequence described by the first-order recurrence relation:
un + 1 = 4un u1 = 0.5.
WORKED EXAMPLE 999
2 Graph these � rst � ve terms. The value of the term is plotted on the y-axis, and the term number is plotted on the x-axis.
un
n0
−20
2 43 5
−16−12−8−4
Val
ue o
f te
rm
Term number
4
232 MATHS QUEST 12 FURTHER MATHEMATICS VCE Units 3 and 4
c05RecurrenceRelations.indd 232 12/09/15 4:08 AM
UNCORRECTED >
UNCORRECTED > 1)
UNCORRECTED 1)
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
0
UNCORRECTED
0
UNCORRECTED
UNCORRECTED
1 3 5 6
UNCORRECTED
1 3 5 61 3 5 6
UNCORRECTED
1 3 5 61 3 5 6
UNCORRECTED
1 3 5 61 3 5 6
UNCORRECTED
1 3 5 61 3 5 6
UNCORRECTED
1 3 5 6
−8
UNCORRECTED
−8−6
UNCORRECTED
−6−4
UNCORRECTED
−4−2
UNCORRECTED
−20
−20
UNCORRECTED
0−2
0
Val
ue o
f te
rm
UNCORRECTED
Val
ue o
f te
rm
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED
An increasing saw pattern occurs
UNCORRECTED
An increasing saw pattern occurs
PAGE Value o
f te
rmPAGE Value
of
term
PROOFS are distinguished by
PROOFS are distinguished by
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
2PROOFS
23PROOFS
34
PROOFS
45
PROOFS5
Val
ue o
f te
rm
PROOFS
Val
ue o
f te
rm
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
THINK WRITE/DRAW
1 Generate the six terms of the sequence. un + 1 = 4un u1 = 0.5u2 = 4u1
= 4 × 0.5= 2
u3 = 4u2
= 4 × 2= 8
u4 = 4u3
= 4 × 8= 32
u5 = 4u4
= 4 × 32= 128
u6 = 4u5
= 4 × 128= 512
2 Graph these terms.
Note: The sixth term is not included in this graph to more clearly illustrate the relationship between the terms.
n
un
20
40
41 3 5
80120160
Val
ue o
f te
rm
Term number
Interpretation of the graph of first-order recurrence relationsStraight or linear
A straight line or linear pattern is given by first-order recurrence relations of the form un + 1 = un + d
Non-linear (exponential)
A non-linear pattern is generated by first-order recurrence relations of the form un + 1 = Run
20
10
41 3 5 6 7 8 9 10 11
203040
Val
ue o
f te
rm
Term number
506070y
x
20
20
41 3 5 6 7 8 9 10 11
406080
Val
ue o
f te
rm
Term number
100120140
y
x
Topic 5 RECURRENCE RELATIONS 233
c05RecurrenceRelations.indd 233 12/09/15 4:08 AM
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED A straight line or linear pattern is
UNCORRECTED A straight line or linear pattern is given by first-order recurrence
UNCORRECTED given by first-order recurrencerelations of the form
UNCORRECTED relations of the form
UNCORRECTED
n
UNCORRECTED n +
UNCORRECTED + d
UNCORRECTED d
N
UNCORRECTED
Non
UNCORRECTED
on-linear (exponential)
UNCORRECTED
-linear (exponential)
UNCORRECTED
UNCORRECTED
UNCORRECTED PAGE
PAGE
PAGE
PAGE Term number
PAGE Term number
Interpretation of the graph of first-order recurrence relations
PAGE Interpretation of the graph of first-order recurrence relations
PAGE
PAGE
A straight line or linear pattern is PAGE
A straight line or linear pattern is
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
nPROOFS
nPROOFS
1 3 5PROOFS
1 3 541 3 54 PROOFS
41 3 54 PROOFS
Term numberPROOFS
Term numberPROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
Starting termEarlier, the need for a starting term to be given to fully de� ne a sequence was stated. As can be seen below, the same pattern but a different starting point gives a different set of numbers.
un + 1 = un + 2 u1 = 3 gives 3, 5, 7, 9, 11, …
un + 1 = un + 2 u1 = 2 gives 2, 4, 6, 8, 10, …
The first five terms of a sequence are plotted on the graph. Write the first-order recurrence relation that defines this sequence.
n
un
20
2
41 3 5
468
Val
ue o
f te
rm
Term number
1012141618
THINK WRITE
1 Read from the graph the � rst � ve terms of the sequence.
The sequence from the graph is:18, 15, 12, 9, 6, …
2 Notice that the graph is linear and there is a common difference of −3 between each term.
un + 1 = un + dCommon difference, d = −3un + 1 = un − 3 (or un + 1 − un = −3)
3 Write your answer including the value of one of the terms (usually the � rst), as well as the rule de� ning the � rst order difference equation.
un + 1 = un − 3 u1 = 18
WORKED EXAMPLE 101010
The first four terms of a sequence are plotted on the graph. Write the first-order recurrence relation that defines this sequence.
n
un
20
2
41 3 5
468
10
Val
ue o
f te
rm
Term number
12141618
WORKED EXAMPLE
234 MATHS QUEST 12 FURTHER MATHEMATICS VCE Units 3 and 4
c05RecurrenceRelations.indd 234 12/09/15 4:08 AM
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED 18, 15, 12, 9, 6, …
UNCORRECTED 18, 15, 12, 9, 6, …
Notice that the graph is linear and there is a
UNCORRECTED Notice that the graph is linear and there is a
3 between each term.
UNCORRECTED 3 between each term.
u
UNCORRECTED u
Write your answer including the value
UNCORRECTED Write your answer including the value of one of the terms (usually the � rst),
UNCORRECTED
of one of the terms (usually the � rst), as well as the rule de� ning the � rst order
UNCORRECTED
as well as the rule de� ning the � rst order
UNCORRECTED
The first four terms of a sequence are plotted on the graph. Write the first-
UNCORRECTED
The first four terms of a sequence are plotted on the graph. Write the first-order recurrence
UNCORRECTED
order recurrence
UNCORRECTED
UNCORRECTED
UNCORRECTED
UNCORRECTED PAGE
PAGE
PAGE WRITE
PAGE WRITE
The sequence from the graph is:PAGE The sequence from the graph is:18, 15, 12, 9, 6, …PAGE
18, 15, 12, 9, 6, …
PROOFS
THINK WRITE
1 Read the terms of the sequence from the graph. The sequence is 2, 4, 8, 16, …
2 The graph is non-linear and there is a common ratio of 2, that is, for the next term, multiply the previous term by 2.
un + 1 = R × un
Common ratio, R = 2un + 1 = 2un
3 De� ne the � rst term. u1 = 2
4 Write your answer. un + 1 = 2un u1 = 2
The first five terms of a sequence are plotted on the graph shown. Which of the following first-order recurrence relations could describe the sequence?
A un + 1 = un + 1 with u1 = 1
B un + 1 = un + 2 with u1 = 1
C un + 1 = 2un with u1 = 1
D un + 1 = un + 1 with u1 = 2
E un + 1 = un + 2 with u1 = 2
THINK WRITE
1 Eliminate the options systematically. Examine the � rst term given by the graph to decide if it is u1 = 1 or u1 = 2.
The coordinates of the � rst point on the graph are (1, 1).The � rst term is u1 = 1.Eliminate options D and E.
2 Observe any pattern between each successive point on the graph.
There is a common difference of 2 or un + 1 = un + 2.
3 Option B gives both the correct pattern and � rst term.
The answer is B.
un
n40
1
2 31 5
2345
Val
ue o
f te
rm
Term number
9876
10
WORKED EXAMPLE 121212
Graphs of first-order recurrence relations1 WE8 On a graph, show the � rst � ve terms of a sequence described by the
� rst-order recurrence relation:un + 1 = un − 1 u1 = −2.
2 On a graph, show the � rst � ve terms of a sequence described by the � rst-order recurrence relation:
un + 1 = un + 3 u1 = 1.
3 WE9 On a graph, show the � rst six terms of a sequence described by the � rst-order recurrence relation:
un + 1 = 3un u1 = 2.
EXERCISE 5.4
PRACTISE
Topic 5 RECURRENCE RELATIONS 235
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UNCORRECTED
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UNCORRECTED WRITE
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Eliminate the options systematically.
UNCORRECTED Eliminate the options systematically. Examine the � rst term given by
UNCORRECTED Examine the � rst term given by the graph to decide if it is
UNCORRECTED the graph to decide if it is u
UNCORRECTED u1
UNCORRECTED
1 =
UNCORRECTED = 1
UNCORRECTED 1
The coordinates of the � rst point on the graph are (1, 1).
UNCORRECTED The coordinates of the � rst point on the graph are (1, 1).
Observe any pattern between each
UNCORRECTED
Observe any pattern between each successive point on the graph.
UNCORRECTED
successive point on the graph.
gives both the correct pattern
UNCORRECTED
gives both the correct pattern and � rst term.
UNCORRECTED
and � rst term.
UNCORRECTED
EXERCISE 5.4
UNCORRECTED
EXERCISE 5.4
UNCORRECTED
PRACTISEUNCORRECTED
PRACTISE
PAGE
Val
ue o
f te
rm
PAGE
Val
ue o
f te
rmPROOFS
PROOFSThe first five terms of a sequence are plotted on the graph shown. Which of
PROOFSThe first five terms of a sequence are plotted on the graph shown. Which of relations could describe the sequence?
PROOFSrelations could describe the sequence?
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
Val
ue o
f te
rmPROOFS
Val
ue o
f te
rm
9
PROOFS98
PROOFS
87PROOFS
76PROOFS
6PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
4 On a graph, show the �rst six terms of a sequence described by the �rst-order recurrence relation:
un+ 1 = 12
un u1 = 36.
5 WE10 The �rst �ve terms of a sequence are plotted on the graph shown. Write the �rst-order recurrence relation that de�nes this sequence.
6 The �rst �ve terms of a sequence are plotted on the graph shown. Write the �rst-order recurrence relation that de�nes this sequence.
7 WE11 The �rst four terms of a sequence are plotted on the graph shown. Write the �rst-order recurrence relation that de�nes this sequence.
8 The �rst four terms of a sequence are plotted on the graph shown. Write the �rst-order recurrence relation that de�nes this sequence.
9 WE12 The �rst �ve terms of a sequence are plotted on the graph shown. Which of the following �rst-order recurrence relations could describe the sequence?
A un + 1 = un + 1 u1 = 2B un + 1 = un + 2 u1 = 2C un + 1 = un + 3 u1 = 1D un + 1 = un + 1 u1 = 1E un + 1 = un + 2 u1 = 1
12
16un
n
8
42
1 2 3 4 5
6
10
14
0
Term number
Val
ue o
f te
rm
12
16182022un
n
8
1 2 3 4 5
6
10
14
0
Term number
Val
ue o
f te
rm
n
un
20
2
41 3 5
468
1012
Val
ue o
f te
rm
Term number
n
un
20
6
41 3 5
1218243036
Val
ue o
f te
rm
Term number
n
un
20
2
41 3 5
468
1012
Val
ue o
f te
rm
Term number
236 MATHS QUEST 12 FURTHER MATHEMATICS VCE Units 3 and 4
c05RecurrenceRelations.indd 236 12/09/15 4:08 AM
UNCORRECTED The �rst four terms of a sequence are plotted
UNCORRECTED The �rst four terms of a sequence are plotted on the graph shown. Write the �rst-order recurrence
UNCORRECTED on the graph shown. Write the �rst-order recurrencerelation that de�nes this sequence.
UNCORRECTED relation that de�nes this sequence.
The �rst four terms of a sequence are plotted on the
UNCORRECTED
The �rst four terms of a sequence are plotted on the graph shown. Write the �rst-order recurrence
UNCORRECTED
graph shown. Write the �rst-order recurrencethat de�nes this sequence.
UNCORRECTED
that de�nes this sequence.
PAGE PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS1 2 3 4 5
PROOFS1 2 3 4 5
PROOFS
PROOFS
PROOFS
PROOFS0
PROOFS0
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFSTerm number
PROOFSTerm number
PROOFS20
PROOFS2022
PROOFS22u
PROOFSun
PROOFSn
PROOFS
PROOFS
PROOFS
Val
ue o
f te
rmPROOFS
Val
ue o
f te
rm
10 The �rst four terms of a sequence are plotted on the graph shown. Which of the following �rst-order recurrence relations could describe the sequence?
A un + 1 = 2un u1 = 2 B un + 1 = 2un u1 = 1C un + 1 = 2un u1 = 0 D un + 1 = 3un u1 = 1E un + 1 = 3un u1 = 2
11 For each of the following, plot the �rst �ve terms of the sequence de�ned by the �rst-order recurrence relation.
a un + 1 = un + 3 u1 = 1 b un + 1 = un + 7 u1 = 5
c un + 1 = un − 3 u1 = 17
12 For each of the following, plot the �rst �ve terms of the sequence de�ned by the �rst-order recurrence relation.
a un + 1 = −2un u1 = −0.5 b un = 0.5un − 1 u1 = 16
c un + 1 = 2.5un u1 = 2
13 For each of the following, plot the �rst four terms of the sequence de�ned by the �rst-order recurrence relation.
a un + 1 = 3un − 4 u1 = −3 b un = 2un − 1 + 0.5 u1 = 2
c un = 2 + 5un − 1 u1 = 0.2
14 For each of the following, plot the �rst four terms of the sequence de�ned by the �rst-order recurrence relation.
a un + 1 = 100 − 3un u1 = 20 b un + 1 = un − 50 u1 = 100
c un = 0.1un − 1 u1 = 10
15 On graphs, show the �rst 5 terms of the sequences de�ned by the following recurrence relations.
a un + 1 = un + 3, u1 = 1, n = 1, 2, 3, …
b un + 1 = un − 3, u1 = 17, n = 1, 2, 3, …
c un + 1 = un − 15, u1 = 75, n = 1, 2, 3, …
d un + 1 = un + 10, u1 = 80, n = 1, 2, 3, …
16 On graphs, show the �rst 4 terms of the sequences de�ned by the following recurrence relations.
a un + 1 = 6un, u1 = 1, n = 1, 2, 3, …
b un + 1 = 3un, u1 = −1, n = 1, 2, 3, …
c un + 1 = 1.5un, u0 = 1, n = 0, 1, 2, 3, …
d un + 1 = 0.5un, u1 = 10, n = 1, 2, 3, …
17 For each of the following graphs, write a �rst-order recurrence relation that de�nes the sequence plotted on the graph.a
n
un
20
4
41 3 5
8121620
Val
ue o
f te
rm
Term number
b
n
un
20
20
41 3 5
406080
100
Val
ue o
f te
rm
Term number
c un
n0 2 41 3 5
−6−4−2
Val
ue o
f te
rm
Term number
2468
CONSOLIDATEn
un
20
2
41 3 5
468
1012
Val
ue o
f te
rm
Term number
Topic 5 RECURRENCE RELATIONS 237
c05RecurrenceRelations.indd 237 12/09/15 4:08 AM
UNCORRECTED On graphs, show the �rst 5 terms of the sequences de�ned by the following
UNCORRECTED On graphs, show the �rst 5 terms of the sequences de�ned by the following relations.
UNCORRECTED relations.
3,
UNCORRECTED 3, u
UNCORRECTED u1
UNCORRECTED 1 =
UNCORRECTED = 1,
UNCORRECTED 1,
−
UNCORRECTED − 3,
UNCORRECTED 3, u
UNCORRECTED u
u
UNCORRECTED un
UNCORRECTED n −
UNCORRECTED − 15,
UNCORRECTED 15,
UNCORRECTED
=
UNCORRECTED
= u
UNCORRECTED
un
UNCORRECTED
n +
UNCORRECTED
+ 10,
UNCORRECTED
10,
On graphs, show the �rst 4 terms of the sequences de�ned by the following
UNCORRECTED
On graphs, show the �rst 4 terms of the sequences de�ned by the following recurrence
UNCORRECTED
recurrence
a
UNCORRECTED
a u
UNCORRECTED
un
UNCORRECTED
n +
UNCORRECTED
+ 1
UNCORRECTED
1
b
UNCORRECTED
b u
UNCORRECTED
uc
UNCORRECTED
c
PAGE For each of the following, plot the �rst four terms of the sequence de�ned by the
PAGE For each of the following, plot the �rst four terms of the sequence de�ned by the
20
PAGE 20 10 PAGE 10
On graphs, show the �rst 5 terms of the sequences de�ned by the following PAGE
On graphs, show the �rst 5 terms of the sequences de�ned by the following
PROOFS1
PROOFS1
For each of the following, plot the �rst �ve terms of the sequence de�ned by the
PROOFSFor each of the following, plot the �rst �ve terms of the sequence de�ned by the
n
PROOFSn −
PROOFS− 1
PROOFS1
PROOFS
For each of the following, plot the �rst four terms of the sequence de�ned by the
PROOFSFor each of the following, plot the �rst four terms of the sequence de�ned by the
u PROOFS
un PROOFS
n =PROOFS
= 2PROOFS
2
18 For each of the following graphs, write a �rst-order recurrence relation that de�nes the sequence plotted on the graph.a
n
un
2
5
410 3 5
10
30
40
Val
ue o
f te
rm
Term number
35
252015
b un
n0 2 41 3 5
−6−4−2
Val
ue o
f te
rm
Term number
2468
c un
n0 21 3 54
−6−4−2
Val
ue o
f te
rm
Term number
2468
19 The �rst �ve terms of a sequence are plotted on the graph.
Which of the following �rst-order recurrence relations could describe the sequence?
A un + 1 = un − 8 u1 = 8B un + 1 = un + 8 u1 = 8C un + 1 = un − 8 u1 = 45D un + 1 = un + 8 u1 = 45E un + 1 = 8un u1 = 45
20 Write the �rst-order recurrence relation that de�nes the sequence plotted on the graph shown.
un
n0 41 3 5
−25−20−15−10−5
5
21 Graphs of the �rst �ve terms of �rst-order recurrence relations are shown below together with the �rst-order recurrence relations. Match the graph with the �rst-order recurrence relation by writing the letter corresponding to the graph together with the number corresponding to the �rst-order recurrence relation.a un
n0 41 32 5
−6Val
ue o
f te
rm
Term number
−4−2
2b
n
un
20
8
41 3 5
16243240
Val
ue o
f te
rm
Term number
c
n
un
20
4
41 3 5
8121620
Val
ue o
f te
rm
Term number
d
n
un
20
4
41 3 5
8121620
Val
ue o
f te
rm
Term number
e
n
un
20
1
41 3 5
2345
Val
ue o
f te
rm
Term number
f
n
un
20
2
41 3 5
468
10
Val
ue o
f te
rm
Term number
n
un
20
10
41 3 5
20304050
Val
ue o
f te
rm
Term number
MASTER
238 MATHS QUEST 12 FURTHER MATHEMATICS VCE Units 3 and 4
c05RecurrenceRelations.indd 238 12/09/15 4:08 AM
UNCORRECTED 0
UNCORRECTED 0−
UNCORRECTED −−5
UNCORRECTED −5
Graphs of the �rst �ve terms of �rst-order recurrence
UNCORRECTED
Graphs of the �rst �ve terms of �rst-order recurrencetogether with the �rst-order recurrence
UNCORRECTED
together with the �rst-order recurrenceorder recurrence
UNCORRECTED
order recurrencewith the number corresponding to the �rst-order recurrence
UNCORRECTED
with the number corresponding to the �rst-order recurrencea
UNCORRECTED
a u
UNCORRECTED
u
Val
ue o
f te
rm
UNCORRECTED
Val
ue o
f te
rm
PAGE Write the �rst-order recurrence relation that de�nes the sequence plotted on the
PAGE Write the �rst-order recurrence relation that de�nes the sequence plotted on the
PAGE
PAGE
PAGE
PAGE u PAGE un PAGE n
5 PAGE
5 PAGE
PAGE
PAGE
PAGE
PAGE
PAGE
PAGE
PAGE PROOFS
PROOFS
PROOFS
PROOFSTerm number
PROOFSTerm number
PROOFS
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PROOFSu
PROOFSun
PROOFSn
30
PROOFS3040
PROOFS4050
PROOFS50
Val
ue o
f te
rm
PROOFS
Val
ue o
f te
rm
PROOFS
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PROOFS
i un + 1 = un + 12
u1 = 8 ii un + 1 = un − 2 u1 = 1
iii un + 1 = 2un u1 = 1 iv un + 1 = 12
un u1 = 16
v un + 1 = 2un u1 = 2 vi un + 1 = un − 12
u1 = 4
22 The �rst few terms of four sequences are plotted on the graphs below. Match the following �rst-order recurrence relations with the graphs.
i un + 1 = un + 2, u1 = 1, n = 1, 2, 3, …
ii un + 1 = 3un, u1 = 1, n = 1, 2, 3, …
iii un + 1 = −4un, u1 = −5, n = 1, 2, 3, …
iv un + 1 = un − 4, u1 = 1, n = 1, 2, 3, …
a un
n0 2 43 5
−50−40−30−20−10
1020
−60−70−80
b
n
un
20
5
41 3 5
1015202530
c un
n0 41 3 5
−20−15−10−5
510
d
n
un
20
2
41 3 5
468
10
Topic 5 RECURRENCE RELATIONS 239
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UNCORRECTED 1 3 51 3 5
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ONLINE ONLY 5.5 Review www.jacplus.com.au
The Maths Quest Review is available in a customisable format for you to demonstrate your knowledge of this topic.
The Review contains:• Multiple-choice questions — providing you with the
opportunity to practise answering questions using CAS technology
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A summary of the key points covered in this topic is also available as a digital document.
REVIEW QUESTIONSDownload the Review questions document from the links found in the Resources section of your eBookPLUS.
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InteractivitiesA comprehensive set of relevant interactivities to bring dif� cult mathematical concepts to life can be found in the Resources section of your eBookPLUS.
240 MATHS QUEST 12 FURTHER MATHEMATICS VCE Units 3 and 4
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UNCORRECTED to bring dif� cult mathematical concepts to life to bring dif� cult mathematical concepts to life
UNCORRECTED to bring dif� cult mathematical concepts to life can be found in the Resources section of your
UNCORRECTED can be found in the Resources section of your
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PAGE studyON is an interactive and highly visual online
PAGE studyON is an interactive and highly visual online tool that helps you to clearly identify strengths PAGE tool that helps you to clearly identify strengths and weaknesses prior to your exams. You can PAGE
and weaknesses prior to your exams. You can then con� dently target areas of greatest need, PAGE
then con� dently target areas of greatest need, PAGE
PAGE PROOFS
PROOFS
PROOFS
PROOFS
PROOFSDownload the Review questions document from
PROOFSDownload the Review questions document from the links found in the Resources section of your
PROOFSthe links found in the Resources section of your
PROOFS
PROOFS
5 AnswersEXERCISE 5.2
1 a This is a �rst-order recurrence relation.
b This is not a �rst-order recurrence relation.
2 a This is not a �rst-order recurrence relation.
b This is a �rst-order recurrence relation.
3 un = 4un − 1 + 3, u0 = 5
The sequence is 5, 23, 95, 383, 1535.
4 fn + 1 = 5fn − 6, f0 = −2
The sequence is −2, −16, −86, −436, −2186.
5 The second term is 2.
6 The �fth term is 23
.
7 b, c, g, j
8 a 6, 8, 10, 12, 14
b 5, 2, −1, −4, −7
c 23, 24, 25, 26, 27
d 7, −3, −13, −23, −33
9 a 1, 3, 9, 27, 81
b −2, −10, −50, −250, −1250
c 1, −4, 16, −64, 256
d −1, −2, −4, −8, −16
10 a 1, 3, 7, 15, 31
b 5, 13, 37, 109, 325
11 a The first seven terms are 6, −5, 6, −5, 6, −5, 6.
b The first seven terms are 1, 5, 25, 125, 625, 3125, 15 625.
12 C
13 E
14 7
15 −1
16 −18
17 D
18 a un + 1 = 3un, u0 = 14
; 14
, 34
, 2 14
, 6 34
, 20 14
b un + 1 = un + 2000, u0 = 200 000; 200 000, 202 000, 204 000, 206 000, 208 000
c un + 1 = un − 7, u0 = 100; 100, 93, 86, 79, 72
d un + 1 = 2un − 50, u0 = $200; $200, $350, $650, $1250, $2450
EXERCISE 5.31 a un + 1 = un + 3, u1 = 1
b un + 1 = un − 4, u1 = 12
2 a un + 1 = un + 5, u1 = −24
b un + 1 = un + 11, u1 = 55
3 The sequence is −8, −13, −18, −23, −28, ...
The �rst-order recurrence relation is un + 1 = un − 5, u1 = −8.
4 The sequence is 4 12
, 5, 5 12
, 6, 6 12
, ...
The �rst-order recurrence relation is
un +1 = un + 12
, u1 = 4 12
.
5 a un + 1 = 5un, u1 = 1 b un + 1 = −2un, u1 = 4
6 a un + 1 = 5un, u1 = 3
b un +1 = −12
un, u1 = 200
7 a un + 1 = 5un u1 = 2 b un + 1 = 4un u1 = −3
8 a un + 1 = 2un u1 = 3 b un + 1 = 3un u1 = −2
9 a un + 1 = un + 2 u1 = 1
b un + 1 = un + 7 u1 = 3
c un + 1 = un − 7 u1 = 12
d un + 1 = un − 0.5 u1 = 1
e un + 1 = un + 4 u1 = 2
f un + 1 = un + 4 u1 = −2
g un + 1 = un − 5 u1 = 6
h un + 1 = un + 6.5 u1 = 4
10 B
11 a un + 1 = un − 1 u1 = −4
b un + 1 = un + 2 u1 = 3
c un + 1 = un + 3 u1 = −1
d un + 1 = un − 2 u1 = 4
12 A
13 a un + 1 = un − 2, u1 = 12, n = 1, 2, 3, …
b un + 1 = un + 1, u1 = 1, n = 1, 2, 3, …
14 a un + 1 = 2un u1 = 5
b un + 1 = 6un u1 = 1
c un + 1 = −un u1 = −3
d un + 1 = 4un u1 = −3
e un + 1 = 3un u1 = 2
f un + 1 = −un u1 = 5
g un + 1 = 4un u1 = −2
h un + 1 = −3un u1 = 5
15 D
16 a un + 1 = 3un u1 = 2
b un + 1 = 4un u1 = −3
c un + 1 = −un u1 = 0.5
d un + 1 = 5un u1 = 3
e un + 1 = 2un u1 = −5
f un + 1 = −3un u1 = 0.1
Topic 5 RECURRENCE RELATIONS 241
c05RecurrenceRelations.indd 241 12/09/15 4:08 AM
UNCORRECTED 5,
UNCORRECTED 5, 6,
UNCORRECTED 6, −
UNCORRECTED −5,
UNCORRECTED 5, 6
UNCORRECTED 6.
UNCORRECTED .
25,
UNCORRECTED 25, 125,
UNCORRECTED 125, 625,
UNCORRECTED 625, 3125,
UNCORRECTED 3125,
+
UNCORRECTED
+ 1
UNCORRECTED
1 =
UNCORRECTED
= 3
UNCORRECTED
3u
UNCORRECTED
un
UNCORRECTED
n,
UNCORRECTED
, u
UNCORRECTED
u0
UNCORRECTED
0
u UNCORRECTED
unUNCORRECTED
n +UNCORRECTED
+ 1UNCORRECTED
1 =UNCORRECTED
= uUNCORRECTED
unUNCORRECTED
n
204UNCORRECTED
204 000, 206UNCORRECTED
000, 206
PAGE =
PAGE = u
PAGE un
PAGE n
n
PAGE n +
PAGE + 1
PAGE 1 =
PAGE = u
PAGE un
PAGE n −
PAGE − 7
PAGE 7u
PAGE un
PAGE n +
PAGE + 1
PAGE 1 =
PAGE = u
PAGE u
e
PAGE e u
PAGE un
PAGE n +
PAGE + 1
PAGE 1 =
PAGE =
f
PAGE f u
PAGE un
PAGE n
gPAGE g
PROOFSu
PROOFSun
PROOFSn +
PROOFS+ 1
PROOFS1 = −
PROOFS= −2
PROOFS220
PROOFS200
PROOFS0
1
PROOFS1 =
PROOFS= 2
PROOFS2 b
PROOFSb
PROOFS u
PROOFSu1
PROOFS
1 =
PROOFS= 3
PROOFS3
+ PROOFS
+ 2PROOFS
2 uPROOFS
u
7PROOFS
7
17 A
18 a un + 1 = −un, u1 = −4, n = 1, 2, 3, …
b un + 1 = −7un, u1 = 2, n = 1, 2, 3, …
c un + 1 = 3un, u1 = 2, n = 1, 2, 3, …
d un + 1 = −un, u1 = −1, n = 1, 2, 3, …
e un + 1 = 2un, u1 = 5, n = 1, 2, 3, …
19 a un + 1 = un + 6.5, u1 = 4, n = 1, 2, 3, …
b un + 1 = un − 5, u1 = 15, n = 1, 2, 3, …
c un + 1 = un + 4, u1 = 1, n = 1, 2, 3, …
20 a un + 1 = 2un, u1 = 4, n = 1, 2, 3, …
b un + 1 = 4un, u1 = −3, n = 1, 2, 3, …
c un + 1 = −6un, u1 = 2, n = 1, 2, 3, …
d un + 1 = 8un, u1 = −0.1, n = 1, 2, 3, …
e un + 1 = −10un, u1 = 3.5, n = 1, 2, 3, …
EXERCISE 5.41 un
n0 2 41 3 5
−6−5−4−3
Val
ue o
f te
rm
−2
1
−1
2
n
un
20
2
41 3 5
468
Val
ue o
f te
rm
101214
3
n
un
20
100
41 3 5 6
200300400
Val
ue o
f te
rm 500
4
n
un
20
3
41 3 5 6
69
12
Val
ue o
f te
rm
Term number
1518212427303336
5 un + 1 = un − 2 u1 = 14
6 un + 1 = un + 4 u1 = 6
7 un + 1 = 2un u1 = 1.5
8 un + 1 = 2un u1 = 3
9 B
10 B
11 a
n
un
20
4
41 3 5
81216
Val
ue o
f te
rm
Term number
20 b
n
un
20
10
41 3 5
203040
Val
ue o
f te
rm
Term number
50
c
n
un
20
5
41 3 5
101520
Val
ue o
f te
rm
Term number
25
12 a
−12
Term number
un
n0 2 41 5V
alue
of
term
−8−4
48
b
n
un
20
4
41 3 5
81216
Val
ue o
f te
rm
Term number
20
c
n
un
20
20
41 3 5
406080
Val
ue o
f te
rm
Term number
100
13 a
Term number
un
n0 21 5
Val
ue o
f te
rm
−400−500
−300−200−100
100
b
n
un
20
10
41 3 5
203040
Val
ue o
f te
rm
Term number
50
c
n
un
20
100
41 3 5
200300400
Val
ue o
f te
rm
Term number
500
242 MATHS QUEST 12 FURTHER MATHEMATICS VCE Units 3 and 4
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rmPAGE V
alue
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term 100PAGE
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PROOFS1 3 5
PROOFS1 3 5Term number
PROOFSTerm number
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
PROOFS
14 a un
n0 2 41 5
204060
Val
ue o
f te
rm80
140120
160
100
−20
Term number
b un
n0 2 41 3 5Val
ue o
f te
rm
Term number
−25−50
255075
100
c
n
un
20
2
41 3 5
468
Val
ue o
f te
rm
Term number
10
15 a
n
un
20
3
41 3 5
69
1215
b
n
un
20
5
41 3 5
10152025
c
n
un
20
20
41 3 5
406080
d
n
un
20
30
41 3 5
6090
120150
16 a
n
un
20
50
41 3 5
100150200250
b un
n0 41 3 5
−20−25
−15−10−5
5
c
n
un
20
1
41 3 5
23456
d
n
un
20
2
41 3 5
468
10
17 a un + 1 = un + 3 u1 = 2
b un + 1 = un − 20 u1 = 90
c un + 1 = un − 4 u1 = 8
18 a un + 1 = 2un u1 = 5
b un + 1 = −un u1 = 6
c un + 1 = −0.5un u1 = 8
19 C
20 un + 1 = 3un u1 = −1
21 a ii
b v
c iii
d iv
e vi
f i
22 i matches to d.
ii matches to b.
iii matches to a.
iv matches to c.
Topic 5 RECURRENCE RELATIONS 243
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nPROOFS
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1 3 5PROOFS
1 3 541 3 54 PROOFS
41 3 54 PROOFS
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