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Recurrence Relations, Arithmetic and Geometric Sequence Year 12 Mathematics Applications Recurrence Relations & Arithmetic and Geometric Sequence Except where indicated, this content is © Department of Education Western Australia 2020 and released under a Creative Commons CC BY NC licence. Before re-purposing any third party content in this resource refer to the owner of that content for permission. Mathematics Applications - 12 Recurrence Relations & Arithmetic and Geometric Sequences Contents Signposts… ............................................................................................................................. 2 Overview …. ............................................................................................................................ 3 Lesson 1 – Patterns and relationships ....................................................................... 5 Lesson 2 – Patterns and relationships ..................................................................... 12 Lesson 3 – Rules for sequences .................................................................................. 18 Lesson 4 – Difference equations ................................................................................ 25 Lesson 5 – Difference equations ................................................................................ 31 Lesson 6 – Arithmetic sequences ............................................................................... 36 Lesson 7 – Arithmetic sequences ............................................................................... 41 Lesson 8 – Geometric sequences ............................................................................... 48 Lesson 9 – Geometric sequences ............................................................................... 53 Lesson 10 – Application of sequences ..................................................................... 58 Lesson 11 – Exam Practice – Calculator Free ........................................................ 65 Lesson 12 – Exam Practice – Calculator allowed ................................................. 68 Further practice ................................................................................................................. 72 Glossary ............................................................................................................................. 72 Summary ............................................................................................................................. 74 Solutions ............................................................................................................................. 76 © Department of Education Western Australia 2020 1 Arithmetic and Geometric Sequence introduction Mathematics Applications - 12 Signposts Each symbol is a sign to help you. Here is what each one means. Important Information Mark and Correct your work You write an answer or response Use your CAS calculator A point of emphasis Refer to a text book Contact your school teacher (if you can) Check your school about Assessment submission © Department of Education Western Australia 2020 2 Mathematics Applications - 12 Recurrence Relations & Arithmetic and Geometric Sequences Overview This Booklet contains approximately 4 weeks of work. You may find it necessary to do ‘homework’ in order to finish it. To guide the pace at which you work through the booklet, refer to the contents page. Space is provided for you to write your solutions in this PDF booklet. If you need more space, then attach a page to the page you are working on. Answers are given to all questions. It is assumed you will use them responsibly, to maximise your learning. You should check your day to day lesson work. Assessments All of your assessments are provided for you separately by your school. Assessments will be either response or investigative. Weightings for assessments are provided by your school. Calculator This course assumes the use of a CAS calculator. Screen displays will appear throughout the booklets to help you with your understanding of the lessons. Further support documents are available. Textbook You are encouraged to use a text for this course. A text will further explain some topics and can provide you with extra practice questions. Online Support Search for a range of online support. © Department of Education Western Australia 2020 3 Arithmetic and Geometric Sequence introduction Mathematics Applications - 12 Content covered in this booklet The syllabus content focused on in this booklet includes: The arithmetic sequence 3.2.1 use recursion to generate an arithmetic sequence 3.2.2 display the terms of an arithmetic sequence in both tabular and graphical form and demonstrate that arithmetic sequences can be used to model linear growth and decay in discrete situations 3.2.3 deduce a rule for the term of a particular arithmetic sequence from the pattern of the terms in anℎ arithmetic sequence, and use this rule to make predictions 3.2.4 use arithmetic sequences to model and analyse practical situations involving linear growth or decay The geometric sequence 3.2.5 use recursion to generate a geometric sequence 3.2.6 display the terms of a geometric sequence in both tabular and graphical form and demonstrate that geometric sequences can be used to model exponential growth and decay in discrete situations 3.2.7 deduce a rule for the term of a particular geometric sequence from the pattern of the terms in theℎ sequence, and use this rule to make predictions 3.2.8 use geometric sequences to model and analyse (numerically, or graphically only) practical problems involving geometric growth Sequences generated by first-order linear recurrence relations 3.2.9 use a general first-order linear recurrence relation to generate the terms of a sequence and to display it in both tabular and graphical form 3.2.10 generate a sequence defined by a first-order linear recurrence relation that gives long term increasing, decreasing or steady-state solutions 3.2.11 use first-order linear recurrence relations to model and analyse (numerically or graphically only) practical problems © Department of Education Western Australia 2020 4 Mathematics Applications - 12 Recurrence Relations & Arithmetic and Geometric Sequences Lesson 1 Patterns and relationships By the end of this lesson you should be able to: • consider situations in which number patterns are generated • make predictions about number sequences based on observed patterns • devise ways of describing patterns of a sequence by a recurrence relation. Patterns and Relationships Patterns and relationships are really what mathematics is all about. Many great discoveries and predictions have been made from patterns being identified and the reasons for them being explored. Fibonacci is one pattern we will explore. The diagram on the left shows how each Fibonacci number is determined by adding the two previous terms. Fibonacci patterns are found extensively in nature, like the Spiral Aloe plant on the right! Image: Call Tree for Fibonacci Number, Wikimedia. Image: J Brew, Aloe polyphilla. Wikimedia , Sequences A number sequence is a set of numbers arranged in a definite order. For example 3, 8, 13, 18, 23,… is a sequence formed by a starting number, 3, and then by adding 5 to generate the next number. As this sequence continues indefinitely, it is called an infinite sequence. © Department of Education Western Australia 2020 5 Arithmetic and Geometric Sequence introduction Mathematics Applications - 12 The sequence 5, 10, 20, 50, 100, 200 is the sequence of values (in cents) of Australian coins. This sequence is a finite sequence because it has a definite number of terms. Look at the sequence 3, 8, 13, 18, 23, … Each number of the sequence is called a term. 3 is the first term i.e. t1 8 is the second term i.e. t2 13 is the third term i.e. t3 etc Note: The general term for a sequence is usually given the symbol tn , but this can be either lower case or upper case (capital letters), so Tn is equally acceptable. The subscript, n, tells us the term number. Skills Development 1.1 1. For the sequence 4, 7, 10, 13, 16, 19, … determine − (a) t4 (b) tt32 + − (c) 3tt34 (d) 2 (tt54) © Department of Education Western Australia 2020 6 Mathematics Applications - 12 Recurrence Relations & Arithmetic and Geometric Sequences 2. For the sequence 1, 3, 6, 10, 15, 21, 28, … determine − (a) T3 (b) 3TT12 − − (c) TT56 (d) TT14(2 T 3) Determining particular terms of a sequence If tn is any particular term then tn +1 is the next term in the sequence (the term one on from tn ) and tn +2 is the term following tn +1 or the term two on from tn Similarly If tn is any particular term then tn −1 is the term before tn or the previous term of the sequence and tn −2 is the term before tn −1 or two terms before tn Using symbols, the sequence looks like this: … , tn −2 , tn −1 , tn , tn +1 , tn +2 , … © Department of Education Western Australia 2020 7 Arithmetic and Geometric Sequence introduction Mathematics Applications - 12 Example Take the sequence 3, 8, 13, 18, 23, … = If we choose tn 13 then find: (a) tn +1 (b) tn +2 (c) tn −1 (d) tn −2 Solution (a) tn +1 (b) tn +2 (c) tn −1 (d) tn −2 = 18 = 23 = 8 = 3 Example Take the sequence 3, 8, 13, 18, 23, … = If we choose tn 18 then find: (a) tn +1 (b) tn +2 (c) tn −1 (d) tn −2 Solution (a) tn +1 (b) tn +2 (c) tn −1 (d) tn −2 = 23 = 28 = 13 = 8 Skills Development 1.2 1. For each of the sequences below: (a) Look at the pattern and find the next two terms. (b) Describe the pattern in words. (c) Complete the statement about the terms. © Department
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