Chapter 4 4Arithmetic sequences Objectives To explore sequences of numbers and their recurrence relations. To use a calculator to generate sequences and display their graphs. To recognise arithmetic sequences, and to find their terms, recurrence relations and numbers of terms. To calculate the sum of the terms in an arithmetic series. To apply arithmetic sequences and series to solving problems. The following are examples of sequences of numbers: a 1, 3, 5, 7, 9,... b 10, 7, 4, 1,,... c 0.6, 1.7,.8,..., 9.4 1 3, 1 9, 1 7, 1 d 81,... e 0.1, 0.11, 0.111, 0.1111, 0.11111,... Each sequence is a list of numbers, with order being important. Sequence c is an example of a finite sequence, and the others are infinite sequences. For some sequences of numbers, we can give a rule for getting from one number to the next: a rule for sequence a is: add a rule for sequence b is: subtract 3 a rule for sequence c is: add 1.1 a rule for sequence d is: multiply by 1 3 The first three sequences are called arithmetic sequences: we add or subtract a fixed amount to get from one number to the next. In this chapter, we begin with an introduction to sequences and then we focus on arithmetic sequences and their applications. We will study geometric sequences (such as sequence d) in Chapter 14. Knowledge check See the online test of required knowledge, with links to revision lessons.
4A Introduction to sequences 143 4A Introduction to sequences The numbers of a sequence are called its terms. The nth term of a sequence is denoted by the symbol t n. So the first term is t 1, the 1th term is t 1, and so on. Recurrence relations A sequence may be defined by a rule which enables each subsequent term to be found from the previous term. This type of rule is called a recurrence relation, a recursive formula or an iterative rule. For example: The sequence 1, 3, 5, 7, 9,... may be defined by t 1 = 1 and t n = t n 1 +. The sequence 1 3, 1 9, 1 7, 1 81,... may be defined by t 1 = 1 3 and t n = 1 3 t n 1. Example 1 Use the recurrence relation to find the first four terms of the sequence t 1 = 3, t n = t n 1 + 5 t 1 = 3 t = t 1 + 5 = 8 t 3 = t + 5 = 13 t 4 = t 3 + 5 = 18 The first four terms are 3, 8, 13, 18. Example Find the recurrence relation for the following sequence: 9, 3, 1, 1 3,... 3 = 1 3 9 i.e. t = 1 3 t 1 1 = 1 3 ( 3) i.e. t 3 = 1 3 t The sequence is defined by t 1 = 9 and t n = 1 3 t n 1. A sequence may also be defined explicitly by a rule that is stated in terms of n. For example: The rule t n = n defines the sequence t 1 =, t = 4, t 3 = 6, t 4 = 8,... The rule t n = n 1 defines the sequence t 1 = 1, t =, t 3 = 4, t 4 = 8,... The sequence 1, 3, 5, 7, 9,... can be defined by t n = n 1. The sequence t 1 = 1 3, t n = 1 3 t n 1 can be defined by t n = 1 3 n.
144 Chapter 4: Arithmetic sequences Example 3 Find the first four terms of the sequence defined by the rule t n = n + 3. t 1 = (1) + 3 = 5 t = () + 3 = 7 t 3 = (3) + 3 = 9 t 4 = (4) + 3 = 11 The first four terms are 5, 7, 9, 11. Example 4 Find a rule for the nth term of the sequence 1, 4, 9, 16 in terms of n. t 1 = 1 = 1 t = 4 = t 3 = 9 = 3 t 4 = 16 = 4 t n = n Example 5 At a particular school, the number of students studying Mathematical Methods increases each year. There are presently 40 students studying Mathematical Methods. a Set up the recurrence relation if the number is increasing by five students each year. b Write down an expression for t n in terms of n for the recurrence relation found in part a. c Find the number of students expected to be studying Mathematical Methods at the school in five years time. a t n = t n 1 + 5 b t 1 = 40 t = t 1 + 5 = 45 = 40 + 1 5 t 3 = t + 5 = 50 = 40 + 5 Therefore t n = 40 + (n 1) 5 = 35 + 5n c Five years from now implies n = 6: t 6 = 40 + 5 5 = 65 Sixty-five students will be studying Mathematical Methods in five years.
4A Introduction to sequences 145 Example 6 The height of a sand dune is increasing by 10% each year. It is currently 4 m high. a Set up the recurrence relation that describes the height of the sand dune. b Write down an expression for t n in terms of n for the recurrence relation found in part a. c Find the height of the sand dune seven years from now. a t n = t n 1 1.1 b t 1 = 4 t = 4 1.1 = 4.4 t 3 = 4 (1.1) = 4.84 Therefore t n = 4 (1.1) n 1 c Seven years from now implies n = 8: t 8 = 4 (1.1) 7 7.795 The sand dune will be 7.795 m high in seven years. Using a calculator with explicitly defined sequences Example 7 Use a calculator to generate the first 10 terms of the sequence of numbers defined by the rule t n = 3 + 4n Using the TI-Nspire CX non-cas Sequences defined in terms of n can be investigated in a Calculator application. To generate the first 10 terms of the sequence defined by the rule t n = 3 + 4n, complete as shown. Note: Assigning (storing) the resulting list as tn enables the sequence to be graphed. If preferred, the variable tn can be entered as t n using the subscript template, which is accessed via t.
146 Chapter 4: Arithmetic sequences Using the Casio To generate the first 10 terms of the sequence defined by the rule t n = 3 + 4n: Press MENU 8 to select Recursion mode. Select a n as the recursion type: F3 F1 Enter the rule a n = 3 + 4n: 3 + 4 F1 EXE Select Set F5 to adjust the Table Settings. Set the table to start at 1 and end at 10: 1 EXE 1 0 EXE EXIT Select Table F6. Use the cursor key to scroll down the table. Using a calculator with recursively defined sequences Example 8 Use a calculator to generate the sequence defined by the recurrence relation t n = t n 1 + 3, t 1 = 1 and plot the graph of the sequence against n. Using the TI-Nspire CX non-cas In a Lists & Spreadsheet page, name the first two lists n and tn respectively. Enter 1 in cell A1 and enter 1 in cell B1. Note: If preferred, the variable tn can be entered as t n using the subscript template, which is accessed via t.
4A Introduction to sequences 147 Enter = a1 + 1 in cell A and enter = b1 + 3 in cell B. Highlight the cells A and B using shift and the arrows. Use menu > Data > Fill and arrow down to row 10. Press enter to populate the lists. To graph the sequence, insert a Graphs page ( ctrl I > Add Graphs). Graph the sequence as a scatter plot using menu > Graph Entry/Edit > Scatter Plot. Enter the list variables as n and tn in their respective fields. Set an appropriate window using menu > Window/Zoom > Zoom Data. Note: It is possible to see the coordinates of the points: menu > Trace > Graph Trace. The scatter plot can also be graphed in a Data & Statistics page. Alternatively, the sequence can be graphed directly in the sequence plotter ( menu > Graph Entry/Edit > Sequence > Sequence). Enter the rule u1(n) = u1(n 1) + 3 and the initial value 1. Change nstep to 10. Set an appropriate window using menu > Window/Zoom > Zoom Fit. Use ctrl T to show a table of values. Using the Casio Press MENU 8 to select Recursion mode. Select a n+1 as the recursion type: F3 F
148 Chapter 4: Arithmetic sequences 4A Enter the rule a n+1 = a n + 3: Example 1 Example F + 3 EXE Note: The recurrence relation t n = t n 1 + 3 is equivalent to t n+1 = t n + 3. Go to Table Setting F5. Set the table to start at 1 and end at 10; set the initial value a 1 = 1: 1 EXE 1 0 EXE F 1 EXE EXIT To view the graph, select Table F6 and then Graph-Plot F6. To return to the table, press EXIT. Section summary A sequence may be defined by a rule which enables each subsequent term to be found from the previous term. This type of rule is called a recurrence relation and we say that the sequence has been defined recursively. For example: The sequence 1, 3, 5, 7, 9,... is defined by t 1 = 1 and t n = t n 1 +. The sequence 1 3, 1 9, 1 7, 1 81,... is defined by t 1 = 1 3 and t n = 1 3 t n 1. Exercise 4A 1 In each of the following, a recursive definition for a sequence is given. List the first five terms. a t 1 = 3, t n = t n 1 + 4 b t 1 = 5, t n = 3t n 1 + 4 d t 1 = 1, t n = t n 1 + e t n+1 = t n + t n 1, t 1 = 1, t = 3 For each of the following sequences, find the recurrence relation: c t 1 = 1, t n = 5t n 1 a 3, 6, 9, 1,... b 1,, 4, 8,... c 3, 6, 1, 4,... d 4, 7, 10, 13,... e 4, 9, 14, 19,... Example 3 3 Each of the following is a rule for a sequence. In each case, find t 1, t, t 3, t 4. a t n = 1 b t n = n + 1 c t n = n d t n = n n e t n = 3n + f t n = ( 1) n n 3 g t n = n + 1 h t n = 3 n 1
4A 4A Introduction to sequences 149 Example 4 4 For each of the following sequences, find a possible rule for t n in terms of n: a 3, 6, 9, 1,... b 1,, 4, 8,... c 1, 1 4, 1 9, 1 16,... Example 5 Example 6 d 3, 6, 1, 4,... e 4, 7, 10, 13,... f 4, 9, 14, 19,... 5 Consider a sequence for which t n = 3n + 1. Find t n+1 and t n. 6 7 Hamish collects NRL cards. He currently has 15 and he adds three to his collection every week. a Set up the recurrence relation that will generate the number of cards Hamish has in any given week. b Write down an expression for t n in terms of n for the recurrence relation from part a. c Find the number of cards Hamish should have after another 1 weeks. Isobel can swim 100 m in 94.3 s. She aims to reduce her time by 4% each week. a Set up the recurrence relation that generates Isobel s time for the 100 m in any given week. b Write down an expression for t n in terms of n for the recurrence relation from part a. c Find the time in which Isobel expects to be able to complete the 100 m after another 8 weeks. 8 Stephen is a sheep farmer with a flock of 100 sheep. He wishes to increase the size of his flock by both breeding and buying new stock. He estimates that 80% of his sheep will produce one lamb each year and he intends to buy 0 sheep to add to the flock each year. Assuming that no sheep die: a Write the recurrence relation for the expected number of sheep at the end of each year. (Let t 0 = 100.) b Calculate the number of sheep at the end of each of the first five years. 9 Alison invests $000 at the beginning of the year. At the beginning of each of the following years, she puts a further $400 into the account. Compound interest of 6% p.a. is paid on the investment at the end of each year. a Write down the amount of money in the account at the end of each of the first three years. Example 7 10 b Set up a recurrence relation to generate the sequence for the investment. (Let t 1 be the amount in the account at the end of the first year.) c With a calculator or spreadsheet, use the recurrence relation to find the amount in the account after 10 years. For each of the following, use a calculator to find the first six terms of the sequence and plot the graph of these terms against n: a c t n = 3n t n = n b d t n = 5 n t n = 6 n CF
150 Chapter 4: Arithmetic sequences 4A Example 8 11 For each of the following, use a calculator to find the first six terms of the sequence and plot the graph of these terms against n: a t n = (t n 1 ), t 1 = 1.1 b t n = 3 t n 1, t 1 = 7 c t n = t n 1 + 5, t 1 = 1 1 a For a sequence for which t n = n 1, find t 1, t and t 3. b d t n = 4 t n 1, t 1 = 3 For a sequence for which u n = 1 (n n) + 1, find u 1, u and u 3. c What do you notice? d Find t 4 and u 4. 13 Assume that S n = an + bn, for constants a and b. Find S 1, S, S 3 and S n+1 S n. 14 For the sequence defined by t 1 = 1 and t n+1 = 1 (t n + ), find t, t 3 and t 4. tn The terms of this sequence are successive rational approximations of a real number. Can you recognise the number? 15 The Fibonacci sequence is defined by t 1 = 1, t = 1 and t n+ = t n+1 + t n for n 1. Use the rule to find t 3, t 4 and t 5. Show that t n+ = t n + t n 1 for all n. 4B Arithmetic sequences A sequence in which each successive term is found by adding a fixed amount to the previous term is called an arithmetic sequence. That is, an arithmetic sequence has a recurrence relation of the form t n = t n 1 + d, where d is a constant. For example:, 5, 8, 11, 14, 17,... is an arithmetic sequence. The nth term of an arithmetic sequence is given by t n = t 1 + (n 1)d where d is the common difference between successive terms, that is, d = t k t k 1, for all k > 1. We often denote the first term by a and write this formula as t n = a + (n 1)d Note: In an arithmetic sequence, the nth term t n is a linear function of n. Example 9 Find the 10th term of the arithmetic sequence 4, 1,, 5,.... a = 4, d = 3 t n = a + (n 1)d t 10 = 4 + (10 1) 3 = 3 CU
4B Arithmetic sequences 151 Example 10 If 41 is the nth term in the arithmetic sequence 4, 1,, 5,..., find the value of n. a = 4, d = 3 t n = a + (n 1)d = 41 4 + (n 1) 3 = 41 3(n 1) = 45 n 1 = 15 n = 16 Hence 41 is the 16th term of the sequence. Example 11 The 1th term of an arithmetic sequence is 9 and the 5th term is 100. Find a and d, and hence find the 8th term. An arithmetic sequence has rule t n = a + (n 1)d Since the 1th term is 9, we have 9 = a + 11d (1) Since the 5th term is 100, we have 100 = a + 4d () To find a and d, we solve the two equations simultaneously. Subtract (1) from (): 91 = 13d d = 7 From (1), we have Therefore 9 = a + 11(7) a = 68 t 8 = a + 7d = 68 + 7 7 = 19 The 8th term of the sequence is 19.
15 Chapter 4: Arithmetic sequences Example 1 A national park has a series of huts along one of its mountain trails. The first hut is 5 km from the start of the trail, the second is 8 km from the start, the third 11 km and so on. a How far from the start of the trail is the sixth hut? b How far is it from the sixth hut to the twelfth hut? The distances of the huts from the start of the trail form an arithmetic sequence with a = 5 and d = 3. a For the sixth hut: t 6 = a + 5d = 5 + 5 3 = 0 The sixth hut is 0 km from the start of the trail. Simple interest b For the twelfth hut: t 1 = a + 11d = 5 + 11 3 = 38 The distance from the sixth hut to the twelfth hut is t 1 t 6 = 38 0 = 18 km. One application of arithmetic sequences is simple interest. Simple interest is always calculated on the amount originally invested (called the principal). For example, assume that $1000 is invested at a simple interest rate of 10% per annum. Then the amount of interest earned each year is 10%($1000) = $100. 1 After 1 year, the investment will have grown to $1000 + $100 = $1100. After years, the investment will have grown to $1000 + $100 = $100. 3 After 3 years, the investment will have grown to $1000 + 3 $100 = $1300. The value of the investment after n years will be $1000 + n $100. Example 13 Yasmin invests $100 for five years earning 5% per annum simple interest. What is the value of her investment at the end of the five years? Let A n be the value after n years. Then A n = 100 + n 5%(100) = 100 + 60n So the value after five years is A 5 = 100 + 60 5 = 1500 The value of her investment at the end of the five years is $1500.
4B 4B Arithmetic sequences 153 Simple interest Suppose that $P is invested at a simple interest rate of R% per annum. Then the value of the investment after n years, $A n, is given by A n = P + np R 100 Arithmetic mean The arithmetic mean of two numbers a and b is defined as a + b. If the numbers a, c, b are consecutive terms of an arithmetic sequence, then c a = b c c = a + b c = a + b That is, the middle term c is the arithmetic mean of a and b. Section summary An arithmetic sequence has a recurrence relation of the form t n = t n 1 + d, where d is a constant. Each successive term is found by adding a fixed amount to the previous term. For example:, 5, 8, 11,... The nth term of an arithmetic sequence is given by t n = a + (n 1)d where a is the first term and d is the common difference between successive terms, that is, d = t k t k 1, for all k > 1. Simple interest Suppose that $P is invested at a simple interest rate of R% per annum. Then the value of the investment after n years, $A n, is given by A n = P + np R 100. Exercise 4B Example 9 1 For the arithmetic sequence where t n = a + (n 1)d, find the first four terms given that: a a a = 0, d = b a = 3, d = 5 c a = d = 6 d a = 11, d = If an arithmetic sequence has a first term of 5 and a common difference of 3, find the 13th term. b If an arithmetic sequence has a first term of 1 and a common difference of 4, find the 10th term. c For the arithmetic sequence with a = 5 and d =.5, find the ninth term. d For the arithmetic sequence with a = 6 and d =, find the fifth term.
154 Chapter 4: Arithmetic sequences 4B Example 10 Example 11 Example 1 3 Find the rule of the arithmetic sequence whose first few terms are: a 3, 7, 11 b 3, 1, 5 c 1, 3, 7, 11 d, 0, 4 5 In each of the following, t n is the nth term of an arithmetic sequence: a If 54 is the nth term in the sequence 6, 10, 14, 18,..., find the value of n. b If 16 is the nth term in the sequence 5,, 1, 4,..., find the value of n. c Find n if t 1 = 16, t = 13 and t n = 41. d Find n if t 1 = 7, t = 11 and t n = 7. For an arithmetic sequence with fourth term 7 and thirtieth term 85, find the values of a and d, and hence find the seventh term. 6 If an arithmetic sequence has t 3 = 18 and t 6 = 486, find the rule for the sequence, i.e. find t n. 7 For the arithmetic sequence with t 7 = 0.6 and t 1 = 0.4, find t 0. 8 The number of laps that a swimmer swims each week follows an arithmetic sequence. In the 5th week she swims 4 laps and in the 10th week she swims 39 laps. How many laps does she swim in the 15th week? 10 9 For an arithmetic sequence, find t 6 if t 10 = 31 and t 0 = 61. A small company producing wallets plans an increase in output. In the first week it produces 80 wallets. The number of wallets produced each week is to be increased by 8 per week until the weekly number produced reaches 1000. a How many wallets are produced in the 50th week? b In which week does the production reach 1000? 11 An amphitheatre has 5 seats in row A, 8 seats in row B, 31 seats in row C, and so on. a How many seats are there in row P? b How many seats are there in row X? c Which row has 40 seats? 1 The number of people who go to see a movie over a period of a week follows an arithmetic sequence. On the first day only three people go to the movie, but on the sixth day 98 people go. Find the rule for the sequence and hence determine how many attend on the seventh day. Example 13 13 Eva invests $5000 at 4% per annum simple interest. Calculate the amount of interest paid each year and the value of Eva s investment after 10 years. 14 Naveen invested $300 at a simple interest rate of 6% p.a. After how many years will his investment have grown to $4160? 15 Wenny s initial investment of $000 grows to $3600 after investing it for 10 years. What was the rate per annum for the simple interest she earned? CF
4B 4C Arithmetic series 155 16 Five thousand dollars is invested at 6% p.a. simple interest. a Find the total interest for 8 years. b Find the total amount after 8 years. 17 An arithmetic sequence contains 10 terms. If the first is 4 and the tenth is 30, what is the eighth term? 18 The number of goals kicked by an AFL team in the first six games of a season follows an arithmetic sequence. If the team kicked 5 goals in the first game and 15 in the sixth, how many did they kick in each of the other four games? 19 The first term of an arithmetic sequence is a and the mth term is 0. Find the rule for t n for this sequence. 0 Find the arithmetic mean of: a 8 and 15 b 1 and 79 1 Find x if 3x is the arithmetic mean of 5x + 1 and 11. If a, 4a 4 and 8a 13 are successive terms of an arithmetic sequence, find a. 3 If t m = n and t n = m, prove that t m+n = 0. (Here t m and t n are the mth and nth terms of an arithmetic sequence.) 4 If a, a and a are consecutive terms of an arithmetic sequence, find a (a 0). 5 Show that there is no infinite arithmetic sequence whose terms are all prime numbers. 4C Arithmetic series The sum of the terms in a sequence is called a series. If the sequence is arithmetic, then the series is called an arithmetic series. The symbol S n is used to denote the sum of the first n terms of a sequence. That is, S n = a + (a + d) + (a + d) + + ( a + (n 1)d ) Writing this sum in reverse order, we have S n = ( a + (n 1)d ) + ( a + (n )d ) + + (a + d) + a Adding these two expressions together gives S n = n ( a + (n 1)d ) Therefore S n = n ( a + (n 1)d ) Since the last term l = t n = a + (n 1)d, we can also write S n = n ( ) a + l CF CU
156 Chapter 4: Arithmetic sequences Example 14 For the arithmetic sequence, 5, 8, 11,..., calculate the sum of the first 14 terms. a =, d = 3, n = 14 S n = n ( ) a + (n 1)d S 14 = 14 Example 15 ( + 13 3 ) = 301 For the arithmetic sequence 7, 3, 19, 15,..., 33, find: a the number of terms b the sum of the terms. a a = 7, d = 4, l = t n = 33 b a = 7, l = t n = 33, n = 16 t n = a + (n 1)d 33 = 7 + (n 1)( 4) 60 = (n 1)( 4) 15 = n 1 n = 16 There are 16 terms in the sequence. Example 16 For the arithmetic sequence 3, 6, 9, 1,..., calculate: S n = n ( ) a + l S 16 = 16 = 48 ( 7 33 ) The sum of the terms is 48. a the sum of the first 5 terms b the number of terms in the series if S n = 1395. a a = 3, d = 3, n = 5 b a = 3, d = 3, S n = 1395 S n = n S 5 = 5 = 975 ( a + (n 1)d ) ( (3) + (4)(3) ) S n = n ( ) a + (n 1)d = 1395 n ( ) (3) + (n 1)(3) = 1395 n(6 + 3n 3) = 790 3n + 3n = 790 3n + 3n 790 = 0 n + n 930 = 0 (n 30)(n + 31) = 0 Therefore n = 30, since n > 0. Hence there are 30 terms in the series.
4C Arithmetic series 157 Example 17 A hardware store sells nails in a range of packet sizes. Packet A contains 50 nails, packet B contains 75 nails, packet C contains 100 nails, and so on. a Find the number of nails in packet J. b Lachlan buys one each of packets A to J. How many nails in total does Lachlan have? c Assuming he buys one of each packet starting at A, how many packets does he need to buy to have a total of 1100 nails? a a = 50, d = 5 b a = 50, d = 5 t n = a + (n 1)d For packet J, we take n = 10: t 10 = 50 + 9 5 = 75 Packet J contains 75 nails. c a = 50, d = 5, S n = 1100 S n = n ( ) a + (n 1)d = 1100 n ( ) (50) + (n 1)(5) = 1100 n(100 + 5n 5) = 00 5n + 75n 00 = 0 n + 3n 88 = 0 (n + 11)(n 8) = 0 S n = n S 10 = 10 = 165 ( a + (n 1)d ) ( 50 + 9 5 ) Packets A to J contain 165 nails. Thus n = 8, since n > 0. If Lachlan buys one each of the first eight packets (A to H), he will have exactly 1100 nails. Example 18 The sum of the first 10 terms of an arithmetic sequence is 48 3 4. If the fourth term is 3 3 4, find the first term and the common difference. S 10 = 10 t 4 = a + 3d = 3 3 4 a + 3d = 15 4 ( a + 9d ) = 48 3 4 10a + 45d = 195 4 (1) ()
158 Chapter 4: Arithmetic sequences 4C Solve equations (1) and () simultaneously: Skillsheet (1) 40: 40a + 10d = 150 () 4: 40a + 180d = 195 60d = 45 d = 3 4 ( 3 Substitute in (1) to obtain a + 3 = 4) 15 4 and therefore a = 3. The first term is 1 1 and the common difference is 3 4. Section summary The sum of the first n terms of an arithmetic sequence S n = a + (a + d) + (a + d) + + ( a + (n 1)d ) is given by S n = n Exercise 4C 1 ( a + (n 1)d ) or S n = n ( ) a + l, where l = tn For each arithmetic sequence, find the specified sum: Example 14 a 8, 13, 18,... find S 1 b 3.5, 1.5, 0.5,... find S 10 c 1, 1, 3,... find S 15 d 4, 1, 6,... find S 8 For the arithmetic sequence, 7, 1,..., write an expression for: a the nth term b the sum of the first n terms. 3 For the arithmetic sequence,, 6,..., write an expression for: a the nth term b the sum of the first n terms. 4 Greg goes fishing every day for a week. On the first day he catches seven fish and each day he catches three more than the previous day. How many fish did he catch in total? 5 Find the sum of the first 16 multiples of 5. 6 Find the sum of all the even numbers between 1 and 99.
4C 4C Arithmetic series 159 Example 15 Example 16 Example 17 7 For the arithmetic sequence 3, 1, 5, 9,..., 49, find: a the number of terms b the sum of the terms. 8 For the arithmetic sequence 4, 0, 16, 1,..., 5, find: a the number of terms b the sum of the terms. 9 For the arithmetic sequence 1,, 7, 5,..., 17, find: a the number of terms 10 b the sum of the terms. For the sequence 4, 8, 1,..., find: a the sum of the first 9 terms b the value of n if S n = 180. 11 There are 110 logs to be put in a pile, with 15 logs in the bottom layer, 14 in the next, 13 in the next, and so on. How many layers will there be? 1 The sum of the first m terms of an arithmetic sequence with first term 5 and common difference 4 is 660. Find m. 13 Evaluate 54 + 48 + 4 + + ( 54). 14 Dora s walking club plans 15 walks for the summer. The first walk is a distance of 6 km, the last walk is a distance of 7 km, and the distances of the walks form an arithmetic sequence. a How far is the 8th walk? b How far does the club plan to walk in the first five walks? c Dora s husband, Alan, can only complete the first n walks. If he walks a total of 73.5 km, how many walks does he complete? d Dora goes away on holiday and misses the 9th, 10th and 11th walks, but completes all other walks. How far does Dora walk in total? 15 Liz has to proofread 500 pages of a new novel. She plans to read 30 pages on the first day and to increase the number of pages she reads by five each day. a How many days will it take her to complete the proofreading? She has only five days to complete the task. She therefore decides to read 50 pages on the first day and to increase the number she reads by a constant amount each day. b By how many should she increase the number of pages she reads each day if she is to meet her deadline? CF
160 Chapter 4: Arithmetic sequences 4C 16 An assembly hall has 50 seats in row A, 54 seats in row B, 58 seats in row C, and so on. That is, there are four more seats in each row. a How many seats are there in row J? b How many seats are there altogether if the back row is row Z? On a particular day, the front four rows are reserved for parents (and there is no other seating for parents). c How many parents can be seated? d How many students can be seated? The hall is extended by adding more rows following the same pattern. e If the final capacity of the hall is 3410, how many rows were added? 17 A new golf club is formed with 40 members in its first year. Each following year, the number of new members exceeds the number of retirements by 15. Each member pays $10 p.a. in membership fees. Calculate the amount received from fees in the first 1 years of the club s existence. Example 18 18 For the arithmetic sequence with t = 1 and S 1 = 18, find a, d, t 6 and S 6. 19 The sum of the first 10 terms of an arithmetic sequence is 10, and the sum of the first 0 terms is 840. Find the sum of the first 30 terms. 0 If t 6 = 16 and t 1 = 8, find S 14. 1 For an arithmetic sequence, find t n if: a t 3 = 6.5 and S 8 = 67 b t 6 = 17 and S 10 = 155 For the sequence with t n = bn, where b is a constant, find: a t n+1 t n b t 1 + t + + t n 3 For a sequence where t n = 15 5n, find t 5 and find the sum of the first 5 terms. 4 An arithmetic sequence has a common difference of d and the sum of the first 0 terms is 5 times the first term. Find the sum of the first 30 terms in terms of d. 5 The sum of the first n terms of a particular sequence is given by S n = 17n 3n. a Find an expression for the sum of the first (n 1) terms. b Find an expression for the nth term of the sequence. c Show that the sequence is arithmetic and find a and d. 6 Three consecutive terms of an arithmetic sequence have a sum of 36 and a product of 148. Find the three terms. 7 Show that the sum of the first n terms of an arithmetic sequence is n times the sum of the two middle terms. CF CU
Chapter 4 review 161 AS Nrich Chapter summary The nth term of a sequence is denoted by t n. A recurrence relation enables each subsequent term to be found from previous terms. A sequence specified in this way is said to be defined recursively. e.g. t 1 = 1, t n = t n 1 + A sequence may also be defined by a rule that is stated in terms of n. e.g. t n = n 1 Arithmetic sequences An arithmetic sequence has a recurrence relation of the form t n = t n 1 + d where d is a constant. Each successive term is found by adding a fixed amount to the previous term. For example:, 5, 8, 11,... The nth term of an arithmetic sequence is given by t n = a + (n 1)d where a is the first term and d is the common difference between successive terms, that is, d = t k t k 1, for all k > 1. Simple interest Suppose that $P is invested at a simple interest rate of R% per annum. Then the value of the investment after n years, $A n, is given by A n = P + np R 100. Arithmetic series The sum of the terms in an arithmetic sequence is called an arithmetic series. The sum of the first n terms of an arithmetic sequence is given by S n = n ( ) a + (n 1)d or S n = n ) a + l, where l = tn ( Technology-free questions 1 Find the first six terms of the following sequences: a t 1 = 3, t n = t n 1 4 b t 1 = 5, t n = t n 1 + Find the first six terms of the following sequences: a t n = n b t n = 3n + 3 For the arithmetic sequence 1, 3, 5,..., write an expression for: a the nth term b the sum of the first n terms. 4 For the arithmetic sequence 1, 3, 7,..., write an expression for: a the nth term b the sum of the first n terms. Review
16 Chapter 4: Arithmetic sequences Review 5 The 4th term of an arithmetic sequence is 19 and the 7th term is 43. Find the 0th term. 6 For an arithmetic sequence with t 5 = 0.35 and t 9 = 0.15, find t 14. 7 For an arithmetic sequence with t 6 = 4 and t 14 = 6, find S 10. 8 For the arithmetic sequence 5,, 9,..., find the value of n such that S n = 40. 9 Consider the arithmetic sequence 100, 97, 94,.... a Write an expression for t n. b Find the values of n such that t n < 0. c Find the values of n such that S n < 0. 10 How many terms of the arithmetic sequence 6, 9, 1,... must be taken for their sum to equal 81? 11 The 4th term of an arithmetic sequence is 3 and the nd term is 6. Find the sum of the first 5 terms. 1 Consider the arithmetic sequence 15, 17, 19,.... If the sum of the first 3n terms is double the sum of the first n terms, find the value of n. 13 a How many numbers between 100 and 500 are divisible by 1? b Find the sum of these numbers. Multiple-choice questions 1 The first three terms of the sequence defined by the rule t n = 3n + are A 1,, 3 B, 4, 6 C 5, 7, 9 D 5, 8, 11 E 5, 8, 10 If t 1 = 3 and t n+1 = t n + 3, then t 4 is A 4 B 1 C 9 D 15 E 14 3 For the arithmetic sequence 10, 8, 6,..., we have t 10 = A 8 B 10 4 For the arithmetic sequence 10, 8, 6,..., we have S 10 = A 10 B 0 C 1 D 10 E 8 C 10 D 0 5 If 58 is the nth term of the arithmetic sequence 8, 13, 18,..., then n = A 1 B 11 C 10 D 5 E 3 6 Which of the following is not an arithmetic sequence? E 0 A 11,, 8, 19,... B 4, 7, 10, 13,... C 57, 51, 45, 39,... D 3, 5, 7, 9,... E, 8, 14, 0,... CF
Chapter 4 review 163 7 If the general term of an arithmetic sequence is t n = 7 3n, then the sum of the first seven terms of the sequence is A 56 B 35 C 4 D 10 E 34 8 If an arithmetic sequence has third term 1 and sixth term 30, then its tenth term is A 48 B 50 C 54 D 60 E 101 9 The value of 1 + 5 + 9 + + 37 is A 5 B 76 C 190 D 380 E 41 10 A manufacturer produces 000 dolls in the first year, and increases production by 100 dolls each subsequent year. The total number of dolls produced in eight years is A 800 B 18 600 C 18 700 D 18 800 E 18 900 11 The 50th term of the sequence 5,, 9, 16,... is A 35 B 345 C 343 D 338 E 331 1 Dimitry earned $30 000 in his first year of employment. In the following years his salary was increased by $800 each year. What was the total amount that Dimitry had earned by the end of his fifteenth year of employment? A $461 00 B $504 00 C $534 000 D $554 000 E $576 000 Extended-response questions 1 A do-it-yourself picture-framing kit is available in various sizes. Size 1 contains 0.8 m of moulding, size contains 1.5 m, size 3 contains. m, and so on. a Form the sequence of lengths of moulding. b Is the sequence of lengths of moulding an arithmetic sequence? c Find the length of moulding contained in the largest kit, size 1. A firm proposes to sell coated seeds in packs containing the following number of seeds: 50, 75, 100, 15,.... a Is this an arithmetic sequence? b Find a formula for the nth term. c Find the number of seeds in the 5th size packet. 3 A number of power poles are to be placed in a straight line between two towns, A and B, which are 3 km apart. The first is placed 5 km from town A, and the last is placed 3 km from town B. The poles are placed so that the intervals starting from town A and finishing at town B are 5, 5 d, 5 d, 5 3d,..., 5 6d, 3 There are seven poles. How far is the fifth pole from town A, and how far is it from town B? CF Review
164 Chapter 4: Arithmetic sequences Review 4 A new electronic desktop telephone exchange, for use in large organisations, is available in various sizes. Size 1 can handle 0 internal lines. Size can handle 36 internal lines. Size 3 can handle 5 internal lines. a Continue the sequence up to T 8. b Write down a formula for T n in terms of n. Size 4 can handle 68 internal lines, and so on. Size n can handle T n internal lines. c A customer needs an exchange to handle 196 lines. Is there a version of the desktop exchange which will just do this? If so, which size is it? If not, which is the next largest size? 5 A firm makes nylon thread in the following deniers (thicknesses):, 9, 16, 3, 30,... a Find the denier number, D n, of the firm s nth thread in order of increasing thickness. A request came in for some very heavy 191 denier thread, but this turned out to be one stage beyond the thickest thread made by the firm. b How many different thicknesses does the firm make? 6 A new house appears to be slipping down a hillside. The first year it slipped 4 mm, the second year 16 mm, and the third year 8 mm. If it goes on like this, how far will it slip during the 40th year? 7 Anna sends 16 Christmas cards the first year, 4 the second year, 3 the next year, and so on. How many Christmas cards will she have sent altogether after 10 years if she keeps increasing the number sent each year in the same way? 8 In its first month of operation, a cement factory produces 4000 tonnes of cement. In each successive month, production rises by 50 tonnes per month. This growth in production is illustrated for the first five months in the table shown. Month number (n) 1 3 4 5 Cement produced (tonnes) 4000 450 4500 4750 5000 a Find an expression, in terms of n, for the amount of cement produced in the nth month. b Find an expression, in terms of n, for the total amount of cement produced in the first n months. c In which month is the amount of cement produced 950 tonnes? d In month m, the amount of cement produced is T tonnes. Find m in terms of T. e The total amount of cement produced in the first p months is 5 750 tonnes. Find the value of p. CU CF CU