General Mathematics Topic 4: Number Patterns and Recursion This topic includes: the concept of a sequence as a function use of a first- order linear recurrence relation to generate the terms of a number sequence tabular and graphical display of sequences. Key knowledge the concept of sequence as a function and its recursive specification the use of a first- order linear recurrence relation to generate the terms of a number sequence including the special cases of arithmetic and geometric sequences; and the rule for the nth term, tn, of an arithmetic sequence and a geometric sequence and their evaluation the use of a first- order linear recurrence relation to model linear growth and decay, including the rule for evaluating the term after n periods of linear growth or decay the use of a first- order linear recurrence relation to model geometric growth and decay, including the use of the rule for evaluating the term after n periods of geometric growth or decay Key skills use a given recurrence relation to generate an arithmetic or a geometric sequence, deduce the rule for the nth term from the recursion relation and evaluate use a recurrence relation to model and analyse practical situations involving discrete linear and geometric growth or decay formulate the recurrence relation to generate the Fibonacci sequence and use this sequence to model and analyse practical situations. For this topic ALL QUESTIONS are included in these notes at the end of each section.
GENERAL MATHEMATICS 2018 Table of Contents TOPIC 4: NUMBER PATTERNS AND RECURSION... 1 A. SEQUENCES... 3 RANDOMLY GENERATED SEQUENCES... 3 RULE BASED SEQUENCES... 3 NAMING THE TERMS IN A SEQUENCE... 3 ARITHMETIC SEQUENCES... 4 Example 1... 4 USING REPEATED ADDITION ON A CAS CALCULATOR TO GENERATE A SEQUENCE... 5 GRAPHS OF ARITHMETIC SEQUENCES... 5 EXERCISE 1.... 6 B. USING A RECURRENCE RELATION TO GENERATE AND ANALYSE AN ARITHMETIC SEQUENCE... 9 GENERATING THE TERMS OF A FIRST- ORDER RECURRENCE RELATIONS... 9 Example 2... 9 THE IMPORTANCE OF THE STARTING TERM... 10 FINDING OTHER TERMS IN A RECURRENCE RELATION (A GENERAL RULE)... 10 FINDING THE nth TERM IN AN ARITHMETIC SEQUENCE... 10 Example 3- Finding the nth term of an arithmetic sequence... 10 EXERCISE 2.... 11 C. GEOMETRIC SEQUENCES... 13 THE COMMON RATIO, R... 13 Example 4... 13 IDENTIFYING GEOMETRIC SEQUENCES... 14 Example 5... 14 USING REPEATED MULTIPLICATION ON A CAS CALCULATOR TO GENERATE A GEOMETRIC SEQUENCE... 14 GRAPHS OF GEOMETRIC SEQUENCES... 15 EXERCISE 3... 15 D. USING A RECURRENCE RELATION TO GENERATE AND ANALYSE A GEOMETRIC SEQUENCE... 17 Example 6... 17 GENERAL FORM OF THE RECURRENCE RELATION FOR A GEOMETRIC SEQUENCE... 18 Example 7... 18 EXERCISE 4.... 19 E. MODELLING PRACTICAL SITUATIONS (LINEAR GROWTH AND DECAY)... 21 Example 4:... 21 Depreciating assets... 22 CHAPTER SUMMARY... 23 Page 2
NUMBER PATTERNS AND RECURSION A. Sequences A sequence is a list of numbers in a particular order. The numbers or items in a sequence are called the terms of the sequence. They may be generated randomly or by a rule. Randomly generated sequences Recording the numbers obtained while tossing a die would give a randomly generated sequence, such as: 3, 1, 2, 2, 6, 4, 3,... Because there is no pattern in the sequence there is no way of predicting the next term. Consequently, random sequences are of no relevance to this topic and will NOT be considered. Rule based Sequences Writing down odd numbers starting at 1 would result in a sequence generated by a rule: 1, 3, 5, 7, 9, 11, 13,... There is a rule that allows us to state the next term in the sequence. For example: add 2 to the current odd number to find the term after 13, just add 2 to 13, to get 13 + 2 = 15. The group of three dots ( ) at the end of the sequence is called an ellipsis. An ellipsis is used to show that the sequence continues. In this topic, we will look at sequences that can be generated by a rule. Naming the terms in a sequence The symbols V 0, V 1, V 2, are used as labels or names for the first, second and third terms in the sequence. In the labels V 0, V 1, V 2 the numbers 0, 1, 2 are called subscripts. The subscripts tell us the position of each term in the sequence. So, V 10 is just a name for the term in the sequence NOT the value of the term. e.g. 1, 3, 5, 7, 9, 11 Term 1 2 3 4 5 6 n n=0 n=1 n=2 n=3 n=4 n=5 V n V 0 V 1 V 2 V 3 V 4 V 5 Term 1 2 3 4 5 6 n n=0 n=1 n=2 n=3 n=4 n=5 V n V 0=1 V 1=3 V 2=5 V 3=7 V 4=9 V 5=11 Page 3
GENERAL MATHEMATICS 2018 Arithmetic sequences Sequences that are generated by adding or subtracting a fixed amount to the previous term are called arithmetic sequences. The fixed amount we add or subtract to form an arithmetic sequence recursively is called the common difference. The symbol d is often used to represent the common difference. If a sequence is known to be arithmetic, the common difference can be calculated by simply subtracting any pair of successive terms. If a sequence is not known to be arithmetic BUT is found to have a common difference then the sequence is arithmetic. Common Difference, d In an arithmetic sequence, the fixed number added to (or subtracted from) each term to make the next term is called the common difference, where: d = any term the previous term d = V 8 V 9 d = V : V 8 d = V ; V : and so on, For example, the common difference for the arithmetic sequence 20, 25, 30, is: or d = V 8 V 9 = 25 20 = 5 d = V : V 8 = 30 25 = 5 etc Example 1 Finding the common difference in an arithmetic sequence Find the common difference for the following arithmetic sequences and use it to find the 3 rd term in the sequence: a) 2,5,8,... b) 25,23,21,... Page 4
Using repeated addition on a CAS calculator to generate a sequence NUMBER PATTERNS AND RECURSION As we have seen, a recursive rule based on repeated addition, such as to find the next term, add 6, is a quick and easy way of generating the next few terms of a sequence. However, it becomes tedious to do by hand if we want to find, say, the next 20 terms. Fortunately, your CAS calculator can semi- automate the process. Graphs of arithmetic sequences If we plot the values of the terms of an arithmetic sequence (V n ) against their number (n) or position in the sequence, we will find that the points lie on a straight line. An upward slope indicates regular growth and a downward slope reveals decay at a constant rate. A line with positive slope rises from left to right. A negative slope falls from left to right. Page 5
GENERAL MATHEMATICS 2018 Exercise 1. 1. Find the required terms from the sequence: 6, 11, 16, 21, 26, 31... a) V 1 b) V 3 c) V 4 d) V 5 e) V 2 f) V 0 2. For each sequence state the value of the named terms: i) V 1 ii) V 3 iii) V 0 a) 6, 10, 14, 18,... b) 2, 8, 32, 128,... c) 29, 22, 15, 8,... d) 96, 48, 24, 12,... 3. Find out which of the sequences below is arithmetic. Give the common difference for each sequence that is arithmetic. a) 8, 11, 14, 17,... b) 7, 15, 22, 30,... c) 11, 7, 3, 1,... d) 12, 9, 6, 3,... e) 16,8,4,2,... f) 1, 1, 1, 1,... 4. For each of these arithmetic sequences, find the common difference and the 5th term. a) 5, 11, 17, 23,... b) 17, 13, 9, 5,... c) 11, 15, 19, 23,... d) 8, 4, 0, 4,... e) 35, 30, 25, 20,... f) 1.5, 2, 2.5, 3,... 5. Give the next two terms in each of these arithmetic sequences. a) 17, 23, 29, 35,... b) 14, 11, 8, 5,... c) 2, 1.5, 1.0, 0.5,... d) 27, 35, 43, 51,... e) 33, 21, 9, 3,... f) 0.8, 1.1, 1.4, 1.7,... Page 6
NUMBER PATTERNS AND RECURSION 6. Using your CAS calculator: a) Generate the first six terms of the arithmetic sequence: 1, 6, 11...and write down V 5. b) Generate the first 12 terms of the arithmetic sequence: 45, 43, 41... and write down V 12. c) Generate the first 10 terms of the arithmetic sequence: 15, 14, 13,... and write down V 10. d) Generate the first 15 terms of the arithmetic sequence: 0, 3, 6,... and write down V 15. 7. The number of sticks used to make the hexagonal patterns opposite form the arithmetic sequence: 6, 11, 16,... a) Write the common difference for this sequence. b) Using your CAS calculator, determine the number of matches needed to form: i) pattern 6 ii) pattern10 8. After one week of business Fumbles Restaurant had 320 wine glasses. After two weeks, they only had 305 wine glasses. On average 15 glasses are broken each week. Use your CAS calculator, to determine how many weeks it takes at that breakage rate for there to be only 200 glasses left? 9. Elizabeth stored 350 songs on her phone in the first month. In each month that followed she stored 35 more songs. Using your CAS calculator: a) determine the number of songs she had stored after each of the first 4 months b) determine the number of songs she had stored by the end of the first year. Page 7
GENERAL MATHEMATICS 2018 10. a) Graphing the terms of the arithmetic sequence 4, 7, 10,... i. Construct a table showing the term number (n) and its value (t n ) for the first five terms in the sequence. ii. Use the table to plot the graph. iii. Describe the graph. b) Graphing the terms of the arithmetic sequence 9, 7, 5,... i. Construct a table showing the term number (n) and its value (t n ) for the first five terms in the sequence. ii.use the table to plot the graph. iii.describe the graph. Page 8
NUMBER PATTERNS AND RECURSION B. Using a Recurrence Relation to generate and analyse an arithmetic sequence Generating the terms of a first- order recurrence relations A first- order recurrence relation relates a term in a sequence to the previous term in the same sequence. To generate the terms in the sequence, only the initial term is required. A recurrence relation is a mathematical rule that we can use to generate a sequence. It has two parts: 1. a starting point: the value of one of the terms in the sequence 2. a rule that can be used to generate successive terms in the sequence. For example, in words, a recursion rule that can be used to generate the sequence: 10, 15, 20,... can be written as follows: 1. Start with 10. 2. To obtain the next term, add 5 to the current term and repeat the process. A more compact way of communicating this information is to translate this rule into symbolic form. We do this by defining a subscripted variable. Here we will use the variable V n, but the V can be replaced by any letter of the alphabet. Let V n be the term in the sequence after n iterations*. Using this definition, we now proceed to translate our rule written in words into a mathematical rule. Starting value (n=0) Rule for generating the next term V 0=10 V n+1=v n+5 Next term =current term +5 Recurrence relation (two parts: starting value plus rule) V 0=10 V n+1=v n+5 Starting value rule Note: Because of the way we defined V n, the starting value of n is 0. At the start there have been no applications of the rule. This is the most appropriate starting point for financial modelling. Example 2 For the sequence 2, 7, 12, 17, a) Determine if it is an arithmetic sequence d = any term the previous term d = V 8 V 9 = 7 2 = 5 d = V : V 8 = 12 7 = 5 Yes it is arithmetic b) Hence, if it is an arithmetic sequence, state the common difference hence d = 5 c) State the Recurrence Relation for the sequence Recurrence relation V 9 = 2, V?@8 = V? + d, here d = 5 so: The Recurrence Relation is: V 9 = 2, V?@8 = V? + 5 d) Using your CAS list the first 10 terms of the sequence (hint n = 0 n = 9) * Each time we apply the rule it is called an iteration. Page 9
GENERAL MATHEMATICS 2018 The importance of the Starting Term In the example 2 above, If the same rule is used with a different starting point, it will generate different sets of numbers. Example 2 V I = 2, V?@8 = V? + 5 The first five terms were: 2, 7, 12, 17, 22 If V0 =1 then, they would be: 1, 6, 11, 16, 21 If V0 =3 then, they would be: 3, 8, 13, 18, 23 Here you can clearly see that the effect the value of the starting point has. Hence, a recurrence relation MUST have it s starting value stated at ALL TIMES Finding other Terms in a recurrence relation (A General Rule) We can also use recurrence relations to find previous terms, but we need two pieces of information 1. The rule, in terms of V n+1 and V n 2. The term number and its value. i.e. n=2 and V 2=10 (note if n=0, 1, 2, then n=2 is the 3 rd term) Finding the nth term in an arithmetic sequence In Example 2(a), above, the sequence is 4, 7, 10, 13, where V 9 = 4 and d = 3. Writing this out gives: V 9 = 4 = V 9 + 0 3 = 4 V 1 = V 9 + 3 = [V 9 + 3] = V 9 + 1 3 = 7 V 2 = V 8 + 3 = [V 9 + 3 + 3] = V 9 + 2 3 = 10 V 3 = V : + 3 = [V 9 + 3 + 3] + 3 = V 9 + 3 3 = 13 V 4 = V ; + 3 = [V 9 + 3 + 3 + 3] + 3 = V 9 + 4 3 = 16 We can see that a pattern has emerged, that is: where V? the nth term, V 9 is the starting term, d = common difference n = position number of the term. V? = V 9 + n d, Example 3- Finding the nth term of an arithmetic sequence a) Find t 5, the 5th term in the arithmetic sequence: 21, 18, 15, 12,... b) Find t 10, the 10th term in the arithmetic sequence: 9, 7, 5, Page 10
NUMBER PATTERNS AND RECURSION Exercise 2. 1. a) Generate and graph the first five terms of the sequence defined by the recurrence relation: V 0 = 15, V?@8 = V? + 5 where n 1. b) Calculate the value of the 45 th term in the sequence. 2. a) Generate and graph the first five terms of the sequence defined by the recurrence relation: V 0 = 60, V?@8 = V? 5 where n 1. b) Calculate the value of the 10 th term in the sequence. 3. a) Generate and graph the first five terms of the sequence defined by the recurrence relation: V 9 =15, V?@8 = V n + 35 where n 1. b) Calculate the value of the 15 th term in the sequence. Page 11
GENERAL MATHEMATICS 2018 4. The Llama shapes have been made using blocks. Llama 0 Llama 1 Llama 2 Let B n be the number of blocks used to make the nth Llama shape. The number of blocks used to make each Llama shape is generated by the recurrence relation: B 9 = 7, B?@8 = B? + 4 a) Count and record the number of blocks used to make the first, second and third Llama shapes. b) Use the recurrence relation for B n to generate the first five terms of the sequence of perimeters for these shapes. c) Use a rule to calculate the number of blocks needed to make the Llama 8 shape. 5. The BBQ shapes have been made using blocks, each with a side length of 1 unit. BBQ 0 BBQ 1 BBQ 2 The perimeter of each BBQ shape can be found by counting the sides of the blocks around the outside of the shape. Let P n be the perimeter of the nth BBQ shape. The perimeters for this sequence of BBQ shapes is generated by the recurrence relation: P 9 = 16, P?@8 = P n + 6 a) Count and record the perimeters of the first, second and third BBQ shapes. b) Use the recurrence relation for P n to generate the first four terms of this sequence of perimeters. c) Draw the fourth BBQ shape, find its perimeter and check if the recurrence relation correctly predicted the perimeter. d) Use the rule for the nth term for this sequence to predict the perimeter of the 10th BBQ shape (P 89 ). Page 12
NUMBER PATTERNS AND RECURSION C. Geometric sequences The common ratio, r In a geometric sequence, each new term is made by multiplying the previous term by a fixed number called the common ratio, r. This repeating or recurring process is another example of a sequence generated by recursion. In the sequence: each new term is made by multiplying the previous term by 3. The common ratio is 3. In the sequence: 64 32 16 8 4 each new term is made by halving the previous term. In this sequence, we are multiplying each term by 8, which is equivalent to dividing by 2. The common ratio is 8. New terms in a geometric sequence : : V 9, V 8, V ;, V Y, are made by multiplying the previous term by the common ratio, r. Common Ratio, r In a geometric sequence, the common ratio, r, is found by dividing the next term by the current term. current term Common Ratio, r = previous term = V 8 = V : = V ; = V 9 V 8 V : Note: we will only consider values of r > 0 (consider what happens if r < 0) Example 4 Find the common ratio in each of the following geometric sequences. a) 3, 12, 48, 192,... b) 81, 27, 9, 3, Page 13
GENERAL MATHEMATICS 2018 Identifying geometric sequences To identify a sequence as a geometric sequence, it is necessary to find the ratio between multiple pairs of successive terms. If they are common (the same), then it is a geometric sequence. Example 5 Which of the following sequences are geometric sequences? a) 2, 10, 50, 250,... b) 3, 6, 18, 36, Using repeated multiplication on a CAS calculator to generate a geometric sequence As we have seen, using a recursive rule based on repeated multiplication, such as to find the next term, multiply by 2, is a quick and easy way of generating the next few terms of an geometric sequence. It would be tedious to find the next 50 terms. Fortunately, your CAS calculator can semi- automate the process of performing multiple repeated multiplications and do this very quickly. Page 14
NUMBER PATTERNS AND RECURSION Graphs of geometric sequences In contrast with the straight- line graph of an arithmetic sequence, the values of a geometric sequence lie along a curve. Graphing the values of a sequence is a valuable tool for understanding applications involving growth and decay. In the graph above the sequence 2, 4, 8, 16, 32, 64, 128, 256, is an example of geometric growth where r = 2. In the graph above the sequence 256, 128, 64, 32, 16, 8, 4, 2, is an example of geometric decay where r = 1 2 Graphs of Geometric Sequences Graphs of Geometric Sequences (for r > 0, i. e. r is positive) increasing when r is greater than 1, r > 0 decreasing towards zero when r is between 0 and 1, 0 < r < 1. Exercise 3 1. Find out which of the following sequences are geometric. Give the common ratio for each sequence that is geometric. a) 4, 8, 16, 32,... b) 1, 3, 9, 27, c) 5, 10, 15, 20, d) 5, 15, 45, 135, e) 24, 12, 6, 3,... f) 3, 6, 12, 18, g) 4, 8, 12, 16, h) 2, 4, 8, 16 Page 15
GENERAL MATHEMATICS 2018 2. Find the missing terms in each of these geometric sequences. a) 7, 14, 28,,,... b) 3, 15, 75,,, c) 4, 12,,, 324, d),, 20, 40, 80, e) 2,, 32, 128,,... f) 3,, 27,, 243, 729, 3. Use your graphics calculator to generate each sequence and find V e, the sixth term. a) 7, 35, 175,... b) 3, 18, 108, c) 96, 48, 24, d) 4, 28, 196, e) 160, 80, 40,... f) 11, 99, 891, 4. Consider each of the geometric sequences below. i. Find the next two terms. ii. Show the terms in a graph. iii. Describe the graph. a) 3, 6, 12,... b) 8, 4, 2,... Page 16
NUMBER PATTERNS AND RECURSION D. Using a recurrence relation to generate and analyse a geometric sequence Consider the geometric sequence below: 2, 6, 18,... We can continue to generate the terms of this sequence by recognising that it uses the rule: to find the next term multiply the current term by 3 and keep repeating the process. A recurrence relation is a way of expressing this rule in a precise mathematical language. The recurrence relation that generates that sequence 2, 6, 18,... is: V 0 = 2, V?@8 = 3 V n The rule tells us that: the first term is 2, and each subsequent term is equal to the current term multiplied by 3. Understanding this, we proceed to generate the sequence term- by- term as follows: V 9 = 2 V 8 = V 9 3 = 2 3 = 6 V : = V 8 3 = 6 3 = 18 V ; = V : 3 = 18 3 = 54 V Y = V ; 3 = 54 3 = 162 and so on The recurrence relation for generating a geometric sequence is: (the starting term) V 9, V?@8 = V? r, where V? the nth term, V 9 is the starting term, r = common ratio n = position number of the term. Example 6 Generate the recurrence relation for the following geometric sequences a) 4, 8, 16, 32,... b) 1, 3, 9, 27, c) 5, 10, 15, 20, Page 17
GENERAL MATHEMATICS 2018 General form of the recurrence relation for a Geometric Sequence Considering the sequence above: 2, 6, 18,... V 9 = 2 V 8 = V 9 3 = V 9 3 = V 9 3 1 = 6 V 2 = V 8 3 = V 9 3 3 = V 9 3 2 = 18 V 3 = V : 3 = V 9 3 3 3 = V 9 3 3 = 54 V 4 = V ; 3 = V 9 3 3 3 3 = V 9 3 4 = 162 and so on The nth term of geometric sequence can be found by the recurrence relation: (the starting term) V 9, V? = V 9 r?, where V? the nth term, V 9 is the starting term, r = common ratio n = position number of the term. Example 7 a) Generate the first 5 terms of the sequence defined by the recurrence relation: V 9 = 5, V?@8 = 2 V n b) Graph the first 5 terms c) Write down a general recurrence rule to calculate the value of the nth term in the sequence and use it to find V 89. Page 18
NUMBER PATTERNS AND RECURSION Exercise 4. 1. a) Generate first five terms of the geometric sequence defined by the recurrence relation: t 0 = 1000, t n + 1 = 1.1t n. b) Write down a general recurrence rule to calculate the value of the nth term in the sequence and use it to find 13 th term in the sequence correct to two decimal places. 2. a) Generate the first five terms of the geometric sequence defined by the recurrence relation: t 0 = 256, t?@8 = 0.5t n. b) Write down a general recurrence rule to calculate the value of the nth term in the sequence and use it to find 10 th term in the sequence. 3. a) Generate the first five terms of the geometric sequence defined by the recurrence relation: t 1 = 10 000, t?@8 = 1.25t?. Give values to the nearest whole number. b) Write down a general recurrence rule to calculate the value of the nth term in the sequence and use it to find 25 th term in the sequence. Page 19
GENERAL MATHEMATICS 2018 4. A sheet of paper is in the shape of a rectangle. When the sheet is folded once and opened, 2 rectangles are formed either side of the crease. When a sheet is folded twice and opened, 4 rectangles are created, and so on. Note: in the above diagram, n = 0 and hence F 9 are not shown because that is just the unfolded paper Let F n be the number of rectangles created by n folds. The sequence for the number of rectangles created is generated by the recurrence relation: F 0 = 1, F?@8 = 2F n a) Use the recurrence relation for F n to generate the first five terms of the sequence. b) Write down a general recurrence relation for the nth term in the sequence and use it to calculate the number of rectangles after 5 and 10 folds. c) Using your calculator, generate the terms of the sequence to check your answer to b). 6. As a park ranger, Megan has been working on a project to increase the number of rare native orchids in Wilsons Promontory National Park. At the start of the project, a survey found 200 of the orchids in the park. It is assumed from similar projects that the number of orchids will increase by about 18% each year. a) State the first term V 9, and the common ratio r, for the geometric sequence for the number of orchids each year. b) Find a rule for the number of orchids at the start of the nth year. c) How many orchids are predicted in 10 years time? Page 20
NUMBER PATTERNS AND RECURSION E. Modelling practical situations (linear growth and decay) Linear growth and decay is commonly found around the world. They occur when a quantity increases or decreases by the same amount at regular intervals. Everyday examples include the paying of simple interest or the depreciation of the value of a new car by a constant amount each year. An example of linear growth is the investment of money, such as putting it in a savings account where the sum increases over time. An example of linear decay is the money owned to repay a loan, the sum of money owned will decrease over time. Example 4: Jelena puts $5000 into an investment that earns simple interest at a rate of $50 per month. (a) Set up a recurrence relation that represents Jelena s situation as an arithmetic sequence, where Vn+1 is the amount in Jelena s account after n months. (b) Use your equation from part (a) to determine the amount in Jelena s account at the end of each of the first 6 months. n Vn d Vn+1=Vn+d n = 0 V1 = n = 1 V2 = n = 2 V3 = n = 3 V4 = n = 4 V5 = n = 5 V6 = (c) Calculate the amount in Jelena s account at the end of 18 months n = 18, V18 = Page 21
GENERAL MATHEMATICS 2018 Depreciating assets Many items, such as electronic equipment, depreciate over time because of wear and tear. Unit cost depreciation is a way of calculating the value of depreciation according to its use. For example, the value of a cars depreciation is based on how many kilometres it has driven. The value of an item at any given time can be calculated and is referred to as its future value. The write-off value or scrap value of an asset is the point at which the asset is effectively worthless, that is when the value is equal to $0 due to depreciation. Example 5: Loni purchases a new car for $25000 and decides to depreciate it at a rate of $0.20 per km. (a) Set up an equation to determine the value of the car after n km of use. (b) Use your equation from part (a) to determine the future value of the car after it has 7500km on its clock. Page 22
NUMBER PATTERNS AND RECURSION Chapter Summary Sequence Arithmetic sequence Recurrence relation for an arithmetic sequence Linear growth & decay Geometric sequence Recurrence relation for a geometric sequence A sequence is a list of numbers in a particular order. In an arithmetic sequence, each new term is made by adding a fixed number, called the common difference, d, to the previous term. Example: 3, 5, 7, 9,... is made by adding 2 to each term. The common difference, d, is found by taking any term and subtracting its previous term, e.g. V 1 V 0. In our example above, d = 5 3 = 2. A recurrence relation for an arithmetic sequence has the form V 0 = a, V?@8 = V n + d where d = common difference and a = first term. In our example: V 0 = 3, V?@8 = V? + 2 General Rule for finding V n, the nth term in an arithmetic sequence: V n = V 9 + n d To find V? in our example: put n = 10, a = 3, d = 2 V? = 3 + 10 2 = 23 The graph of an arithmetic sequence: " values lie along a straight line Increasing values when d>0 (positive slope) Decreasing values when d<0 (negative slope) An arithmetic sequence can be used to model linear growth (d > 0) or linear decay (d < 0). In a geometric sequence, each term is made by multiplying the previous term by a fixed number, called the common ratio, r. Example: 5, 20, 80, 320,... is made by multiplying each term by 4. The common ratio, r, is found by dividing any term by its previous term, e.g. r = q r q s = q t q r = q u q t In our example: r = q r q s = :9 e = 4 Recurrence relation for a geometric sequence: V 9 = a, V?@8 = r V? where r = common ratio and a = first term. In our example: V 9 = 5, V?@8 = 4 V? General Rule for finding t n, the nth term, in a geometric sequence: V? = V 9 r? where V 9 =first term and r = common ratio. To find V v in our example: put n = 6, a = 5, r = 4 into: V? = 5 4 v = 20,480 The graph of a geometric sequence: Values increase when r > 1 Values decrease towards zero when 0 < r < 1 Page 23