difference equations introduction Generating the terms of a sequence defi ned by a fi rst order difference equation

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1 6 differece equatios areas of STudY Geeratio of the terms of a sequece from a differece equatio, graphical represetatio of such a sequece ad iterpretatio of the graph of the sequece Arithmetic ad geometric sequeces as specific cases of first order liear differece equatios Other first order liear differece equatios used to model chage 6a Geeratig the terms of a sequece defi ed by a fi rst order differece equatio 6b The relatioship betwee arithmetic sequeces ad fi rst order differece equatios 6c The relatioship betwee geometric sequeces ad fi rst order differece equatios 6d Settig up fi rst order differece equatios to represet practical situatios 6e Graphical represetatio of a sequece defi ed by a fi rst order differece equatio 6F Iterpretatio of the graph of fi rst order differece equatios 6G Fiboacci sequeces as secod order differece equatios Settig up ad usig differece equatios to represet practical situatios such as growth models i various cotexts (umerical ad graphical solutio of related equatios) Fiboacci ad related sequeces ad applicatios (umerical ad graphical solutio of related equatios). ebookplus 6a itroductio Digital doc 10 Quick Questios I the previous chapter we examied arithmetic ad geometric patters, examiig such patters with explicit fuctios like t = a + ( - 1) d. Aother approach is to look at how two cosecutive terms i a sequece are related. This approach is more useful i practical applicatios i which the iformatio is provided as follows: The populatio is icreasig by 10% each year, less 00 deaths, with a curret populatio of I the above statemet we are told about the relatioship or chage i populatio from oe year to the ext ad are give a startig term. Geeratig the terms of a sequece defied by a first order differece equatio Cosider the arithmetic sequece: 1, 5, 9, 13, 17,... The commo differece for this sequece is. If t represets the th term of the sequece, the t + 1 represets the ext term; that is, the ( + 1)th term. differece equatios 59

2 We ca defie the arithmetic sequece 1, 5, 9, 13, 17,... with the followig equatio: t + 1 = t + t 1 = 1 The expressio is read as the ext term is the previous term plus, startig at 1. Or, trasposig the above equatio, we get: t t = t 1 = 1. This is read as the differece betwee two cosecutive terms is, startig at 1. This equatio is called a first order differece equatio. It has two mai parts: t + 1 = t + describes the patter i the sequece t 1 = 1 is the first or startig term i the sequece A first order differece equatio defies a relatioship betwee two successive terms of a sequece, for example, betwee: t, the previous term Aother otatio that ca be used is: t - 1, the previous term The first term ca be represeted by either t 0 or t 1. t + 1, the ext term t, the ext term Throughout this chapter we will use either otatio format as short had for ext term, previous term ad first term. Worked Example 1 The followig equatios each defie a sequece. Which of them are first order differece equatios (defiig a relatioship betwee two cosecutive terms)? a t = t t 1 = 3 = 1,, 3,... b t = + = 1,, 3,... c f + 1 = 3f = 1,, 3,... Thik a The equatio cotais the cosecutive terms t ad t - 1 to describe a patter with a kow term. b The equatio cotais oly the t term. There is o t + 1 or t - 1 term. c The equatio cotais the cosecutive terms f ad f + 1 to describe a patter but has o kow first or startig term. Write a This is a first order differece equatio. It has the patter t = t ad a startig or first term of t 1 = 3. b This is ot a first order differece equatio because it does ot describe two cosecutive terms. c This is a icomplete first order differece equatio. It has o first or startig term, so a sequece caot be commeced. Give a fully defied first order differece equatio (patter ad a kow term) we ca geerate the other terms of the sequece. Startig term Earlier, it was stated that a startig term was required to fully defie a sequece. As ca be see below, the same patter with a differet startig poit gives a differet set of umbers. t + 1 = t + t 1 = 3 gives 3, 5, 7, 9, 11,.... t + 1 = t + t 1 = gives,, 6, 8, 10, Maths Quest 1 Further Mathematics for the Casio ClassPad

3 Worked Example Write the first five terms of the sequece defied by the first order differece equatio: t = 3t t 0 = Thik Method 1: Usig the rule Write/DISPLAY 1 Sice we kow the t 0 or startig term, we ca geerate the ext term, t 1, usig the patter: The ext term is 3 the previous term + 5. Now we ca cotiue geeratig the ext term, t, ad so o. t = 3t t 0 = t 1 = 3t = = 11 t = 3t = = 38 t 3 = 3t + 5 = = 119 t = 3t = = 36 3 Write your aswer. The sequece is, 11, 38, 119, 36. Method : Usig a CAS calculator 1 O the Mai scree, the aswer fuctio ca be used to store the previous term. Eter the first term (i this case ) ad press E. To geerate the ext term i the patter, complete the etry lie as: The press E. To geerate the terms of the sequece, cotiue pressig E. 3 Write your aswer. The sequece is, 11, 38, 119, 36. Differece equatios 61

4 Method 3: Usig a spreadsheet 1 Use Excel or the Spreadsheet scree of a CAS calculator. Label colums A ad B as ad t respectively. Eter the first terms ito cells A ad B; they are 1 ad respectively. Complete cell A3 as: = A + 1 Complete cell B3 as: = 3 B + 5 Use the stylus to highlight the cells to fill ad the tap: Edit Fill Rage May terms ca be geerated quickly usig these features. 3 Write your aswer. The sequece is, 11, 38, 119, 36. Worked Example 3 A sequece is defied by the first order differece equatio: t + 1 = t - 3 = 1,, 3,... If the fourth term of the sequece is - 9, that is, t = - 9, the what is the secod term? Thik 1 Traspose the equatio to make the previous term, t, the subject. Use t to fid t 3 by substitutig ito the trasposed equatio. t t Write 3 t = t + 3 = = = 13 6 Maths Quest 1 Further Mathematics for the Casio ClassPad

5 3 Use t 3 to fid t. t t3 + 3 = = = 5 Write your aswer. The secod term, t, is - 5. REMEMBER 1. A first order differece equatio defies a relatioship betwee two successive terms of a sequece, for example, betwee: t, the previous term, ad t + 1, the ext term or t - 1, the previous term, ad t, the ext term.. Remember that otatio such as t - 1 ad t is shorthad for writig the previous term ad the ext term. 3. A first order differece equatio has two mai parts: t + 1 = t + b (where b is a costat) describes the patter i the sequece t 1 = 1 is the first or startig term i the sequece.. First order differece equatios ca be expressed as follows: t + 1 = t + 3 t 0 = 1 is read as the ext term is twice the previous term plus 3, startig at 1 or t t = t 1 = 1 is read as the differece betwee two cosecutive terms is, startig at A startig term is eeded to fully defie a sequece. The same patter with differet startig poits gives differet sets of umbers. t + 1 = t +, t 1 = 3 gives 3, 5, 7, 9, 11,... t + 1 = t +, t 1 = gives,, 6, 8, 10,... Exercise 6a Geeratig the terms of a sequece defied by a first order differece equatio 1 WE 1 Which of the followig equatios are complete first order differece equatios? a t = + b t = t -1-1 t 0 = c t = 1-3t - 1 t 0 = d t - t - 1 = 5 e t = - t - 1 f t = + 1 t 1 = g t = 1 - t - 1 t 0 = 1 h t = a - 1 t = i f + 1 = 3f - 1 j p = p t 0 = 7 Exam tip Studets eed to uderstad that the specificatio of a differece equatio i terms of t ad t - 1 requires a stated value for a startig term, such as t 1. For example t = 0.9 t , t 1 = 5. The value for t 1 = 5 is a essetial requiremet for writig a differece equatio. Most studets omitted this ad could ot score ay marks. [Assessmet report 005] Differece equatios 63

6 WE a Write the first five terms of each of the followig sequeces. i t = t t 0 = 6 ii t = t t 0 = 5 iii t = 1 + t - 1 t 0 = 3 iv t + 1 = t - 10 t 1 = 7 b From your kowledge of chapter 5, write whether the sequeces you have foud i parts i iv are arithmetic or geometric sequeces. 3 a Write the first five terms of each of the followig sequeces. i t = 3t - 1 t 0 = 1 ii t = 5t - 1 t 0 = - iii t = - t - 1 t 0 = 1 iv t + 1 = t t 1 = - 1 b From your kowledge of chapter 5, write whether the sequeces you have foud i parts i iv are arithmetic or geometric sequeces. Write the first five terms of each of the followig sequeces. a t = t t 0 = 1 b t = 3t t 1 = 5 c t = - t t 0 = 6 d t + 1 = 5t - 1 t 1 = 1 5 MC Which of the sequeces below is geerated by the followig first order differece equatio? t = 3t t 0 = a, 3,, 5, 6,... b, 6, 10, 1, 18,... c, 10, 3, 106, 3,... d, 11, 7, 191, 767,... e 6, 10, 1, 18,,... 6 MC Which of the sequeces below is geerated by the followig first order differece equatio? t + 1 = t - 1 t 1 = - 3 a - 3, 5, 9, 17, 33,... b - 3, - 5, - 9, - 17, - 33,... c - 3, 5, - 3, 5, - 3,... d - 3, - 8, - 1, - 6, - 5,... e - 3, - 7, - 15, - 31, - 63,... 7 WE3 A sequece is defied by the first order differece equatio: t + 1 = 3t + 1 = 1,, 3,... If the fourth term is 67, (that is, t = 67) what is the secod term? 8 A sequece is defied by the first order differece equatio: t + 1 = t - 5 = 1,, 3,... If the third term is - 1, (that is, t 3 = - 1) what is the first term? 9 MC A sequece is defied by the first order differece equatio: t + 1 = 5t - 10 = 1,, 3,... If the third term is - 10, the first term is: a 1 6 b 5 6 c 0 d e 10 Write the first order differece equatios for the followig descriptios of a sequece ad geerate the first five terms of the sequece. a The ext term is 3 times the previous term, startig at 1. b Next year s attedace at a motor show is 000 more tha the previous year s attedace, with a first year attedace of c The ext term is the previous term less 7, startig at 100. d The ext day s total sum is double the previous day s sum less 50, with a first day sum of $00. 6 Maths Quest 1 Further Mathematics for the Casio ClassPad

7 6b The relatioship betwee arithmetic sequeces ad first order differece equatios I the ext few exercises, a lik will be made betwee first order differece equatios ad arithmetic ad geometric sequeces studied i chapter 5. Note the variatio i the proumerals used. This is best summarised i a table. Term Arithmetic ad geometric sequece covetio First order differece equatio covetio Worked Example First term a or t 1 t 0 or t 1 Commo differece d b Commo ratio r a This is a ufortuate iterchage of proumerals, but it is most importat to appreciate the traslatio from oe covetio to the other. We kow that a sequece may be defied by a differece equatio. We ca sometimes tell what type of sequece we have by observig the first order differece equatio. Cosider the arithmetic sequece 3, 7, 11, 15, 19,... From chapter 5: d = t - t 1 = t 3 - t = t - t 3 =... d = 7-3 = 11-7 = = + The commo differece is +. This sequece may be defied by the first order differece equatio: t t = t 1 = 3 Rewritig this, we obtai: t + 1 = t + t 1 = 3 A arithmetic sequece with a commo differece of d may be defied by a first order differece equatio of the form: t + 1 = t + b (or t t = b) where b is the commo differece ad for b > 0 it is a icreasig sequece b < 0 it is a decreasig sequece. Which of the followig first order differece equatios defies a arithmetic sequece? a t + 1 = t + t 1 = 3 b t + 1 = t t 1 = 5 c t + 1 = t - 6 t 0 = 11 Thik a Check whether the differece equatio is of the form t + 1 = t + b. b Check whether the differece equatio is of the form t + 1 = t + b. c Check whether the differece equatio is of the form t + 1 = t + b. Write a The first order differece equatio defies a arithmetic sequece with a commo differece of +, a icreasig sequece. b The first order differece equatio does ot defie a arithmetic sequece because the t term has a coefficiet of. c The first order differece equatio defies a arithmetic sequece with a commo differece of - 6, a decreasig sequece. Differece equatios 65

8 Worked Example 5 Express each of the followig arithmetic sequeces as first order differece equatios. a 7, 1, 17,, 7,... b 9, 3, - 3, - 9, - 15,... Thik Write a 1 Write the sequece. a b = t - t 3 b = t 3 - t b = t - t 1 Check for a commo differece. = - 17 = 17-1 = 1-7 = 5 = 5 = 5 3 There is a commo differece of 5 ad the first term is 7. The first order differece equatio is give by t + 1 = t + b t + 1 = t + 5 t 1 = 7 b 1 Write the sequece. b b = t - t 3 b = t 3 - t b = t - t 1 Check for a commo differece. = = = 3-9 = - 6 = - 6 = There is a commo differece of - 6 ad the first term is 9. The first order differece equatio is give by t + 1 = t - 6 t 1 = 9 Worked Example 6 Express the arithmetic sequece defied below as a first order differece equatio. t = = 1,, 3,, 5,... Thik Write 1 Geerate the sequece usig the give rule. t = = 1,, 3,, 5,... t 1 = = 1 = = - 5 t = = = = - 8 t 3 = = 3 = = - 11 t = = = = - 1 There is a commo differece of - 3 ad the first term is - 5. The sequece is - 5, - 8, - 11, - 1,... 3 Write the first order differece equatio. The first order differece equatio is: t + 1 = t - 3 t 1 = Maths Quest 1 Further Mathematics for the Casio ClassPad

9 REMEMBER 1. Proumeral covetios Term Arithmetic ad geometric sequece covetio First order differece equatio covetio First term a or t 1 t 0 or t 1 Commo differece d b Commo ratio r a. The commo differece, b = t - t 1 = t 3 - t = t - t 3 =... A arithmetic sequece with a commo differece of b may be defied by a first order differece equatio of the form: t + 1 = t + b (or t t = b) where b is the commo differece ad for b > 0 it is a icreasig sequece b < 0 it is a decreasig sequece. Exercise 6b The relatioship betwee arithmetic sequeces ad first order differece equatios 1 WE State which of the followig first order differece equatios defie a arithmetic sequece. a t + 1 = 3t t 1 = 6 b t + 1 = t + 3 t 1 = c t + 1 = 3t + 3 t 1 = - 3 d t + 1 = t - t 0 = 5 e t + 1 = t t 0 = 10 f t + 1 = - 3t t 1 = 3 g t + 1 = t - 1 t 1 = 0 h t + 1 = - t t 0 = 1 i t + 1 = t - 3 t 1 = - j t + 1 = t t 1 = 0 WE5 Express each of the followig arithmetic sequeces as first order differece equatios. a 1, 3, 5, 7, 9,... b, 6, 10, 1, 18,... c 3, 10, 17,, 31,... d -,, 6, 10, 1,... e 1, 5, -, - 9, - 16,... f 6, 1, -, - 9, - 1,... g 1, 0.5, 0, - 0.5, - 1,... h, 10.5, 17, 3.5, 30,... 3 MC The arithmetic sequece - 6, - 3, 0, 3, 6,... ca be defied by the first order differece equatio: a t + 1 = t - 3 t 0 = - 6 b t + 1 = t + 3 t 1 = - 6 c t + 1 = 3t t 1 = - 3 d t + 1 = 3t - 1 t 0 = - 3 e t + 1 = 3t t 1 = 3 WE6 Express each of the arithmetic sequeces defied below as first order differece equatios. a t = + 3 = 1,, 3,... b t = - = 1,, 3,... c t = - 10 = 1,, 3,... d t = = 1,, 3,... e t = = 1,, 3,... f t = + 1 = 1,, 3,... g t = 3 - = 1,, 3,... h t = = 1,, 3,... 5 MC The sequece defied by t = - + 3, = 1,, 3,..., ca be defied by the first order differece equatio: a t + 1 = t - t 1 = 1 b t + 1 = - t t 1 = 3 c t + 1 = - t t = 1 d t + 1 = t + 3 t 1 = 1 e t + 1 = - t + 3 t 1 = 1 Differece equatios 67

10 6C The relatioship betwee geometric sequeces ad first order differece equatios Cosider the geometric sequece 1, 3, 9, 7, 81,.... From chapter 5, the commo ratio is give by: r = a = t t3 t = = =... t t t For this sequece a = = 7 3 = 9 =... = + 3 The commo ratio is + 3. This sequece may be defied by the first order differece equatio: 1 t + 1 = 3t t 1 = 1 A geometric sequece with a commo ratio of a may be defied by a first order differece equatio of the form 3 where t + 1 = at a is the commo ratio a > 1 is a icreasig sequece 0 < a < 1 is a decreasig sequece a < 0 is a sequece alteratig betwee positive ad egative values. Worked example 7 Which of the followig first order differece equatios defies a geometric sequece? a t + 1 = t t 1 = 3 b t + 1 = t + t 1 = c t + 1 = 3t - 1 t 1 = d t + 1 = - 5t t 1 = 8 ebookplus Tutorial it-05 Worked example 7 Thik a Check whether the differece equatio is of the form t + 1 = at. b Check whether the differece equatio is of the form t + 1 = at. c Check whether the differece equatio is of the form t + 1 = at. d Check whether the differece equatio is of the form t + 1 = at. WriTe a The first order differece equatio defies a geometric sequece with a commo ratio of, a icreasig sequece. b The first order differece equatio does ot defie a geometric sequece because there is a commo differece of. c The first order differece equatio does ot defie a geometric sequece because of the subtractio of 1. d The first order differece equatio defies a geometric sequece with a commo ratio of - 5. This sequece is alteratig betwee positive ad egative terms. 68 maths Quest 1 Further mathematics for the Casio Classpad

11 Worked Example 8 Express each of the followig geometric sequeces as first order differece equatios. a 1, 5, 5, 15, 65,... b 3, - 6, 1, -, 8... Thik a 1 There is a commo ratio of 5 ad the first term is 1. Write a a = 5 1 = 5 5 = 15 5 =... = + 5 t 1 = 1 Write the first order differece equatio. The first order differece equatio is give by t + 1 = 5t t 1 = 1 b 1 There is a commo ratio of - 6 ad the b a = first term is 3. 3 = 1 6 = 1 =... = - t 1 = 3 Write the first order differece equatio. The first order differece equatio is give by t + 1 = - t t 1 = 3 Worked Example 9 Express each of the geometric sequeces defied below as first order differece equatios. a t = (7) - 1 = 1,, 3,,... b t = - 3() - 1 = 1,, 3,,... Thik Write a 1 We kow from our work with geometric sequeces that their geeral form is t = a(r) - 1, where a represets the first term of the sequece; that is, a = t 1, ad r is the commo ratio. So, i this case t 1 = ad r = 7, where the commo ratio, r, traslates to a = 7. a t = a(r) - 1 t = (7) - 1, = 1,, 3,,... t + 1 = at a = 7, t 1 = 3 Write the first order differece equatio. t + 1 = 7t t 1 = b I this case t 1 = - 3 ad r =. b The first order differece equatio is: t + 1 = t t 1 = - 3 REMEMBER 1. The geometric commo ratio, r, is the proumeral a i first order differece equatios.. r = a = t t3 t = = =... t t t A geometric sequece with a commo ratio of a may be defied by a first order differece equatio of the form: t + 1 = at where a is the commo ratio a > 1 is a icreasig sequece 0 < a < 1 is a decreasig sequece a < 0 is a sequece alteratig betwee positive ad egative values. Differece equatios 69

12 exercise 6C ebookplus Digital doc WorkSHEET 6.1 6d The relatioship betwee geometric sequeces ad first order differece equatios 1 We 7 State which of the followig first order differece equatios defie a geometric sequece. a t + 1 = 3t t 1 = 6 b t + 1 = t + 3 t 1 = c t + 1 = 3t + 3 t 1 = - 3 d t + 1 = t - t 1 = 5 e t + 1 = t t 1 = 10 f t + 1 = - 3t t 1 = 3 g t + 1 = t - 1 t 1 = 0 h t + 1 = - t t 1 = 1 i t + 1 = t - 3 t 1 = - j t + 1 = t t 1 = 0 We8 Express each of the followig geometric sequeces as first order differece equatios. a 5, 10, 0, 0, 80,... b, 6, 18, 5, 16,... c 1, 6, 36, 16, 196,... d 5, - 5, 5, - 5, 5,... e - 3, 3, - 3, 3, - 3,... f -, - 8, - 3, - 18, - 51,... g - 3, - 1, - 8, - 19, - 768,... h 5, - 15, 5, - 135, 05,... 3 mc The geometric sequece -, 6, - 18, 5, - 16,... ca be defied by the first order differece equatio: a t + 1 = - t t 1 = - b t + 1 = - t t 1 = 3 c t + 1 = - 3t t 1 = 3 d t + 1 = - 3t t 1 = - e t + 1 = 3t t 1 = We9 Express each of the geometric sequeces defied below as first order differece equatios. a t = (3) - 1 = 1,, 3,... b t = 3(5) - 1 = 1,, 3,... c t = - 3() - 1 = 1,, 3,... d t = - 5() - 1 = 1,, 3,... e t = 0.5( - 1) - 1 = 1,, 3,... f t = 0.1( - 3) - 1 = 1,, 3,... 5 mc The sequece t = - (1) - 1 = 1,, 3,... ca be defied by the first order differece equatio: a t + 1 = t t 1 = - b t + 1 = t - 1 t 1 = - c t + 1 = t t 1 = 1 d t + 1 = - t + 3 t 1 = - e t + 1 = - t t 1 = - 1 Settig up first order differece equatios to represet practical situatios ebookplus Iteractivity it-0187 Settig up first order differece equatios I practical applicatios we will be preseted with a descriptio of the situatio icludig the patter ad a startig term. We eed to traslate that descriptio to recogise: 1. the first term (t 0 or t 1 ). the curret or previous term (t or t - 1 ) 3. how the ext term (t or t + 1 ) is geerated. The choice of otatio for the first term is defied by the type of situatio. Use t 0 for sequeces that are depedet o time, such as populatio growth ad ivestmet amouts. For example, a populatio starts at 1500 (start = t 0 ). After the first year (1st year = t 1 ) it has grow to 1700, ad after the secod year (d year = t ) it has grow to Use t 1 for most other situatios, such as prizes: first prize t 1 = $1000, secod prize t = $500, third prize t 3 = $ maths Quest 1 Further mathematics for the Casio Classpad

13 As a example, let us look at a very simple situatio. Iitial descriptio: The school populatio is icreasig each year by 50 studets ad the iitial populatio was 00. Below is the descriptio i terms of the populatio for two cosecutive years: Next year s populatio is curret/previous year s populatio plus 50, startig with 00 studets P + 1 = P + 50, P 0 = 00 The first order differece equatio is: P + 1 = P + 50 P 0 = 00 The above approach refers to the two terms as the relatioship betwee the ext (t + 1 ) ad previous (t ) terms. Note: A more appropriate proumeral tha t is usually chose to represet the terms i a sequece, for example, P for populatio. The first order differece equatio ca be stated as: Next term is the previous term plus some defied chage, give the first term. Three types of differece equatios will be closely ivestigated: those describig a arithmetic sequece, a geometric sequece ad a combiatio of both. Type 1 Arithmetic sequece The ext term is the previous term plus a fixed amout or a fixed percetage of a iitial value. t + 1 = t + b where b = the commo differece = fixed amout or percetage of the first term, t 0 or t 1. Worked Example 10 Joh is advised that the rus he scored i his first te iigs i cricket is a patter. The first iigs score was 5, ad each iigs score icreased by 7 rus after that. Write a first order differece equatio to describe this situatio. Thik 1 Defie the terms to be used, for istace, R for rus scored. The first iigs score of 5 is R 1. (The secod iigs is R, the third iigs is R 3, ad so o.) 3 Establish the relatioship betwee the ext ad previous terms. This is a arithmetic sequece because the rus scored i each iigs icrease by the same amout or a commo differece of 7 rus. 5 Write the first order differece equatio. Remember there are two parts: the rule ad the first or startig term. Write Let R = the rus scored i a iigs by Joh, where is the iigs umber. R 1 = 5 Next iigs score is the previous iigs score plus 7 rus. R + 1 = R + 7 R + 1 = R + 7 R 1 = 5 Differece equatios 71

14 Worked example 11 Eri ears % simple iterest per aum o $ that she has ivested. Recall that simple iterest is always calculated o the origial amout. a Write a first order differece equatio to describe this situatio. b Calculate the total value after the third year. ebookplus Tutorial it-053 Worked example 11 Thik WriTe a 1 Defie the terms to be used. a Let A equal the amout Eri s ivestmet would be worth after the th year. The iitial ivestmet was $ at the start of the first moth, that is, A 0. 3 Establish the relatioship betwee the ext ad previous terms. This is a arithmetic sequece because each year the ivestmet icreases by the same amout or a commo differece. 5 Write the first order differece equatio. Remember there are two parts: the rule ad the first or startig term. b 1 Use the first order differece equatio to geerate each of the three terms, A 1, A ad A 3. A 0 = $ The ext term is the previous term plus % of the iitial amout. A + 1 = A + % of $ A + 1 = A A + 1 = A A 0 = b A + 1 = A A 1 = A A = A = = = = A 3 = A = = Write your aswer. After 3 years the total value is $ Type Geometric sequece The ext term is the previous term plus a percetage of the previous term s value. t + 1 = t + % of t or t + 1 = a t where a is the commo ratio. Worked example 1 A bird populatio icreases by 5% each year. If the iitial populatio was 100: a write a first order differece equatio to describe this situatio b calculate the populatio (to the earest whole umber) after 5 years. ebookplus Tutorial it-05 Worked example 1 7 maths Quest 1 Further mathematics for the Casio Classpad

15 Thik a 1 Defie the terms to be used, for istace, P for populatio. The iitial populatio was 100, that is, P 0. 3 Establish the relatioship betwee the ext ad the previous terms. This is a geometric sequece because each year the value icreases by 5%; that is, a = Write the first order differece equatio. Remember there are two parts: the rule ad the first or startig term. a Write/DISPLAY Let P = the populatio after the th year. P 0 = 100 Next year s populatio is previous year s populatio plus 5% of previous year s populatio: P + 1 = P + 5% of P P + 1 = P P = P ( ) = 1.05P P + 1 = 1.05P P 0 = 100 Method 1: Usig the rule b 1 Use the first order differece equatio to geerate each of the six terms, P 0 to P 5. Cotiue this process util P 5 is calculated. 3 Write your aswer, roudig to give a realistic respose; for example, a whole umber of birds. b P + 1 = 1.05P P 0 = 100 P 1 = 1.05P 0 P = 1.05P 1 = = = 160 = 133 P 3 = 1.05P P = 1.05P 3 = = = = P 5 = 1.05P = = After 5 years, the bird populatio is 153, correct to the earest whole umber. Method : Usig a CAS calculator b 1 Ope the Sequece scree by tappig Sequece H. I the Recursive tab, iput the formula usig N. To set the sequece table, tap 8. Complete the table as show to set the start ad ed of the table. Note: a is used i place of t ad the first term is a 0. b Differece equatios 73

16 Ope a Sequece scree. I the Recursive tab, iput the formula usig N. To set the sequece table, tap 8. Complete the table as show to set the start ad ed of the table. To show the list of values of the sequece, ad therefore the populatio after 5 years, tap #. 3 Write your aswer, correct to the earest whole umber. After 5 years, the bird populatio is 153, correct to the earest whole umber. Type 3 Combiatio of arithmetic ad geometric sequeces This is a ew sequece beig ivestigated i this module, where the rule givig the ext term is a combiatio of arithmetic ad geometric sequeces. The ext term is the previous term plus a percetage of the previous term s value plus a fixed amout or a fixed percetage of a iitial value. t + 1 = t + % t + b that is t + 1 = a t + b where a = the commo ratio ad b = the commo differece. Worked Example 13 James is savig for a car. He saves $00 from his pay ad deposits it at the start of each moth ito a accout earig 6% iterest per aum, compoudig mothly ad calculated at the ed of the moth. He opeed the accout o 1 May with a gift from his parets of $100. a Write a first order differece equatio to describe this situatio. b How much would James have o 1 August? 7 Maths Quest 1 Further Mathematics for the Casio ClassPad

17 Thik Write/display a 1 Defie the terms to be used. a Let A = the amout James ivestmet would be worth after the th moth. The iitial ivestmet was $100 at the start of the first moth, that is, A 0. A 0 = $100 3 The ext term is a moth by moth icrease so covert 6% per aum to a icrease per moth. Establish the relatioship betwee the ext ad previous terms. 5 Use the first order differece equatio otatio ad simplify. 6 Write the first order differece equatio. Remember there are two parts; the rule ad the first term. Method 1: Iteratio b 1 Use the differece equatio to geerate the first four terms, A 0 to A 3, for the moths of May, Jue, July ad August. Write your aswer to a appropriate level of accuracy. I this situatio, give the aswer to the earest cet. 6% per aum = 6 % per moth 1 = 0.5% per moth Next moth is previous moth + 0.5% of the previous moth + $00 each moth. A + 1 = A + 0.5% of A + $00 = A ( ) + 00 = 1.005A + 00 A + 1 = 1.005A + 00 A 0 = 100 b A + 1 = 1.005A + 00 A 0 = 100 A 0 = 100 A 1 = 1.005A A = 1.005A = = = = = = A 3 = 1.005A + 00 = = = The total sum James would have o 1 August would be $70.51, correct to the earest cet. Method : Usig a CAS calculator b 1 Ope the Sequece scree. I the Recursive tab, iput the formula usig N. Complete the equatio as: a + 1 = a + 00 a 0 = 100. The set the sequece table by tappig 8. Complete the table as show to set the start ad ed of the b table. Differece equatios 75

18 Make sure the formula box is checked ad the tap ) (this also shows the differece colum). The scree at right should be show. Sice 1 August is 3 terms after the iitial deposit (a 0 ), the value of a 3 is the aswer. 3 Write your aswer to a appropriate level of accuracy. I this situatio, give the aswer to the earest cet. The total sum James would have o 1 August would be $70.51, correct to the earest cet. REMEMBER 1. Type 1 Arithmetic sequece The ext term is the previous term plus a fixed amout or a fixed percetage of a iitial value. t + 1 = t + b where b = the commo differece = fixed amout or % of the first term, t 0 or t 1.. Type Geometric sequece The ext term is the previous term plus a percetage of the previous term s value. t + 1 = t + % of t or t + 1 = a t where a is the commo ratio. 3. Type 3 Combiatio of arithmetic ad geometric sequece The ext term is the previous term plus a percetage of the previous term s value plus a fixed amout or a fixed percetage of a iitial value. t + 1 = t + % t + b t + 1 = a t + b where a = the commo ratio ad b = the commo differece. First term as t 0 Use t 0 as the first term i situatios which are time-depedet, where the startig term is time zero, t 1 is the term after the first time period (for example, the amout after the first week or first year) ad t is the term after the secod time period. 5. First term as t 1 Use t 1 as the first term i situatios which ivolve ordials, such as 1st place, d place ad 3rd place. Aother example is groups, where t 1 is the 1st group, t is the d group ad t 3 is the 3rd group. 76 Maths Quest 1 Further Mathematics for the Casio ClassPad

19 exercise 6d ebookplus Digital doc SkillSHEET 6.1 Chagig a percetage to a decimal Settig up first order differece equatios to represet practical situatios 1 We 10, 11 For each of the situatios below, write a first order differece equatio to describe it. a The first bar o a metal barricade is 50 cetimetres log. Each successive bar is cetimetres loger tha the previous bar. b A machie is programmed to cut legths of rope so that each successive piece is 10 cetimetres loger, startig with a legth of 8 cetimetres. c Paula places $10 from each pay ito her top drawer. She had $50 from Christmas i the drawer to start with. d Water leaks from a tak at the rate of litres per day. The tak iitially held 5000 litres of water. We 1 For each of the situatios below, write a first order differece equatio to describe it ad fid the ukow term. a A tow s populatio icreases by 3% each year. The tow s origial populatio was 600. Fid the populatio after 3 years. b Gary receives a yearly pay icremet of 1.%. His startig salary is $ What is Gary s expected salary after 5 years? c Topsoil at a coastal hillside park is estimated to be eroded at the rate of % per aum. If the estimated amout of topsoil at the park is iitially cubic metres, how much topsoil will be remaiig after years? d A ew hospital icreases the umber of patiets it treats by 1% each year. It treated 3500 i its first year. How may patiets will be treated i the fourth year? 3 We 13 For each of the situatios below, write a first order differece equatio to describe it ad fid the ukow term. a Ats i a at coloy icreased i umber by 7% per week. At the ed of each week 1000 ats are added to the coloy which iitially had ats. How may ats will there be i the coloy after weeks? b A grove of trees loses 1% of trees through evirometal damage each year. Two hudred ew trees are plated each year to cover the losses. The grove bega with 3000 trees. How may trees will there be after 3 years? c Sophie opes a accout ad o the 15th day of each moth deposits $150 ito the accout which ears compoud iterest of 1% per aum, compoudig mothly ad calculated at the ed of the moth. How much will there be i the accout at the ed of the third moth? mc Hele icreases the size of her herb garde by 5% each year. It iitially covered square metres. The first order differece equatio that would describe this is: a A = 0.95A - 1 A 0 = b A = 1.0A - 1 A 0 = c A = 1.05A - 1 A 0 = d A = 1.05A A 0 = 0 e A = 1.05A A 0 = differece equatios 77

20 5 MC Gary rus a fruit-growig busiess. He aims to ear $0 per week more tha i the previous week. I his first week, he ears $160. The first order differece equatio that would describe this is: a A = A A 0 = 160 b A = 160A - 1 A 0 = 0 c A = 1.0A - 1 A 0 = 160 d A = A A 0 = 0 e A = A A 0 = 0 6 MC The umber of people attedig a weight-loss club icreases by 3% each year. Forty members leave the club each year. The club s iitial membership was 1100 statewide. The first order differece equatio that reflects this is: a P = 0.97P P 0 = 1100 b P = 0.97P P 0 = 1100 c P = 1.03P P 0 = 1100 d P = 1.03P P 0 = 1100 e P = 1.0P P 0 = 0 7 MC The umber of people i a coutry tow is decreasig by 5% each year as the youg adults move to the city. A further 0 people die each year. The tow s iitial populatio was 500. The first order differece equatio that reflects this is: a P = 0.95P P 0 = 500 b P = 0.95P P 0 = 500 c P = 1.05P P 0 = 500 d P = 1.05P P 0 = 500 e P = 1.0P P 0 = 0 8 MC The whale populatio i the Souther Pacific Ocea is decreasig by 150 per year. The curret populatio is The first order differece equatio that describes the above is: a P = 0.9P - 1 P 0 = 1500 b P = 1.1P P 0 = 1500 c P + 1 = P P 0 = 1500 d P = P P 0 = 150 e P = P P 0 = The umber of paid-up members of a football club is icreasig by % per week, but the club loses 10 members each week. The club bega with members. a Give the first order differece equatio for the above situatio. b Calculate the size of the membership for each of the first 8 weeks. c I which week will the membership first exceed ? 10 At the local brickworks there are piles of house bricks. The first pile has 000 house bricks. Each pile after the first has 0 fewer house bricks tha the previous pile. a State whether this is a arithmetic or geometric sequece. b Give the first order differece equatio for the above situatio. c Calculate the umber of bricks for each of the first 7 piles. I aother yard, there are piles of pavig bricks. The first pile has 000 pavig bricks; however, the bricks reduce by a rate of 1% for each subsequet pile. d State whether this is a arithmetic or geometric sequece. e Give the first order differece equatio for the above situatio. f Calculate the umber of bricks i the seveth pile of pavig bricks. 78 Maths Quest 1 Further Mathematics for the Casio ClassPad

21 6e Graphical represetatio of a sequece defied by a first order differece equatio Certai quatities i ature ad busiess may chage i a uiform way (formig a patter). This chage may be a icrease, as i the case of t + 1 = t + t 1 = 3, or it may be a decrease, as i the case of t + 1 = t - t 1 = 3. These patters ca be modelled by graphs that, i tur, ca be used to recogise patters i the real world. A graph of the equatio could be draw to represet this situatio. By usig the graph we ca aalyse the situatio to fid, for example, the ext term i the patter. First order differece equatios: t + 1 = t + b (arithmetic patters) The sequeces of a first order differece equatio t + 1 = t + b are distiguished by a straight lie or a costat icrease or decrease. t Term umber A icreasig patter or a positive commo differece gives a upward straight lie. t Term umber A decreasig patter or a egative commo differece gives a dowward straight lie. differece equatios 79

22 Worked example 1 O a graph, show the first five terms of the sequece described by the first order differece equatio: t + 1 = t - 3 t 1 = - 5 Thik WriTe/diSplaY ebookplus Tutorial it-055 Worked example 1 Method 1: Usig a CAS calculator 1 Ope the Sequece scree. I the Recursive tab, tap & ad complete the equatio as: a + 1 = a - 3 a 1 = - 5. The set the sequece table by tappig: 8 ) To chage the widow settigs, tap 6. Set the View Widow scree as show. The tap OK. 3 To view the graph, tap: Graph G-Coect r (to view the full scree graph) Move the cursor over a poit to reveal its coordiates by tappig: Aalysis Trace Use the left/right arrows to trace alog the differet poits. For example, the coordiates (3, - 11) idetify the third term, t 3, as havig a value of maths Quest 1 Further mathematics for the Casio Classpad

23 Method : Usig the rule 1 Geerate the values of each of the five terms of the sequece. Graph these first five terms. The value of the term is plotted o the y-axis, ad the term umber is plotted o the x-axis. t + 1 = t - 3 t 1 = - 5 t = t 1-3 t 3 = t - 3 = = = - 8 = - 11 t = t 3-3 t 5 = t - 3 = = = - 1 = - 17 t Term umber First order differece equatios: t + 1 = at The sequeces of a first order differece equatio t + 1 = at are distiguished by a curved lie or a saw form. t Term umber A icreasig patter or a positive commo ratio greater tha 1 (a > 1) gives a upward curved lie. t Term umber A decreasig patter or a positive fractioal commo ratio (0 < a < 1) gives a dowward curved lie Term umber A icreasig saw patter occurs whe the commo ratio is a egative value less tha - 1 (a < - 1). t Term umber A decreasig saw patter occurs whe the commo ratio is a egative fractio ( - 1 < a < 0). t Differece equatios 81

24 Worked Example 15 O a graph, show the first six terms of the sequece described by the first order differece equatio: t + 1 = t t 1 = 0.5 Thik Method 1: Usig a CAS calculator 1 Ope the Sequece scree. I the Recursive tab, tap & ad complete the equatio as: a + 1 = a a 1 = 0.5. The set the sequece table by tappig: 8 _ Write/display To chage the widow settigs, tap 6. Set: xmi: 0.5 max: 6 ymi: 100 max: 600 The tap OK. To view the graph, tap: Graph G-Coect r (to view the full scree graph) Method : Usig the rule 1 Geerate the six terms of the sequece. t + 1 = t t 1 = 0.5 t = t 1 = 0.5 = t 3 = t = = 8 t = t 3 = 8 = 3 t 5 = t = 3 = 18 t 6 = t 5 = 18 = 51 8 Maths Quest 1 Further Mathematics for the Casio ClassPad

25 Graph these terms. Note: The sixth term is ot icluded i this graph to more clearly illustrate the relatioship betwee the terms. t Term umber Graphical represetatio of first order differece equatios of the form t + 1 = at + b As the patter of first order differece equatios of the form t + 1 = at + b is a combiatio of both arithmetic ad geometric rules, they are primarily distiguished by a curved lie but are more complex i ature tha those give by geometric sequeces. Worked Example 16 O a graph, show the first five terms of the sequece described by the first order differece equatio: t + 1 = 3t - 1 t 1 =. Thik Method 1: Usig a CAS calculator 1 Ope the Sequece scree. I the Recursive tab, tap & ad complete the equatio as: a + 1 = 3a 1 a 1 = The set the sequece table by tappig: 8 ) Write/DISPLAY Differece equatios 83

26 To chage the widow settigs, tap 6. Set: xmi: 0 max: 6 ymi: 0 max: 150 The tap OK. To view the graph, tap: Graph G-Coect r (to view the full scree graph) Method : Usig the rule 1 Geerate the five terms of the sequece. t + 1 = 3t - 1 t 1 = t = 3t 1-1 t 3 = 3t - 1 = 3-1 = = 5 = 1 t = 3t 3-1 t 5 = 3t - 1 = = = 1 = 1 Graph these first five terms. t Term umber REMEMBER 1. A arithmetic patter of a equatio t + 1 = t + b is distiguished by a straight lie or a costat icrease or decrease. t Term umber A icreasig patter or a positive commo differece gives a upward straight lie. t Term umber A decreasig patter or a egative commo differece gives a dowward straight lie. 8 Maths Quest 1 Further Mathematics for the Casio ClassPad

27 . A geometric sequece of a equatio t + 1 = at is distiguished by a curved lie or a saw form. t Term umber A icreasig patter or a positive commo ratio greater tha 1 (a > 1) gives a upward curved lie. t Term umber A icreasig saw patter occurs whe the commo ratio is a egative value less tha - 1 (a < - 1). t Term umber A decreasig patter or a positive fractioal commo ratio (0 < a < 1) gives a dowward curved lie. t Term umber A decreasig saw patter occurs whe the commo ratio is a egative fractio ( - 1 < a < 0). Exercise 6e Graphical represetatio of a sequece defied by a first order differece equatio 1 WE 1 For each of the followig, plot the first five terms of the sequece defied by the first order differece equatio. a t + 1 = t + 3 t 1 = 1 b t + 1 = t + 7 t 1 = 5 c t + 1 = t - 3 t 1 = 17 d t + 1 = t + 5 t 1 = 9 e t + 1 = t + 17 t 1 = 11 f t + 1 = t - 16 t 1 = 90 WE 15 For each of the followig, plot the first five terms of the sequece defied by the first order differece equatio. a t + 1 = 3t t 1 = 1 b t + 1 = t t 1 = 1 c t + 1 = t t 1 = 0.5 d t + 1 = - t t 1 = e t = 0.5t - 1 t 1 = 16 f t + 1 =.5t t 1 = 3 WE 16 For each of the followig, plot the first four terms of the sequece defied by the first order differece equatio. a t + 1 = 3t - 1 t 1 = 1 b t + 1 = 3t - t 1 = - 3 c t + 1 = t + 1 t 1 = 5 d t = t t 1 = e t + 1 = 1 + 3t t 1 = 1 3 f t = + 5t - 1 t 1 = 0. Differece equatios 85

28 ebookplus Digital doc WorkSHEET 6. 6F For each of the followig, plot the first four terms of the sequece defied by the first order differece equatio. a t + 1 = 100-3t t 1 = 0 b t + 1 = t + 50 t 1 = 100 c t + 1 = t - 50 t 1 = 100 d t + 1 = 10t t 1 = 0.1 e t = 0.1t - 1 t 1 = 10 f t = 0.5t t 1 = 30 iterpretatio of the graph of first order differece equatios From the previous exercise, you would have oticed that particular families of graphs were geerated. Straight or liear A straight lie or liear patter is give by first order differece equatios of the form t + 1 = t + b, ad (from the previous chapter) if each pair of terms has a commo differece, it is a arithmetic sequece. o-liear (expoetial) A o-liear patter is geerated by first order differece equatios of the form t + 1 = at, ad (from the previous chapter) if each pair of terms has a commo ratio, it is a geometric sequece. 86 maths Quest 1 Further mathematics for the Casio Classpad

29 Oe other o-liear patter is produced by first order differece equatios of the form t + 1 = at + b: a combiatio of a geometric ad a arithmetic sequece. Startig term Earlier, the eed for a startig term to be give to fully defie a sequece was stated. As ca be see below, the same patter but a differet startig poit gives a differet set of umbers. t + 1 = t + t 1 = 3 gives 3, 5, 7, 9, 11,... t + 1 = t + t 1 = gives,, 6, 8, 10,... Worked Example 17 The first five terms of a sequece are plotted o the graph at right. Write the first order differece equatio that defies this sequece. t Term umber Thik 1 Read from the graph the first five terms of the sequece. Notice that the graph is liear ad there is a commo differece of - 3 betwee each term. 3 Write your aswer icludig the value of oe of the terms (usually the first), as well as the rule defiig the first order differece equatio. Write The sequece from the graph is: 18, 15, 1, 9, 6,... t + 1 = t + b Commo differece, b = - 3 t + 1 = t - 3 (or t t = - 3) t + 1 = t - 3 t 1 = 18 Differece equatios 87

30 Worked example 18 The first four terms of a sequece are plotted o the graph at right. Write the first order differece equatio that defies this sequece. t Term umber Thik WriTe 1 Read the terms of the sequece from the graph. The graph is o-liear ad there is a commo ratio of, that is, for the ext term, multiply the previous term by. 3 Defie the first term. t 0 = The sequece is,, 8, 16,... t + 1 = a t Commo ratio, a = t + 1 = t Write your aswer. t + 1 = t t 0 = Worked example 19 mc The first five terms of a sequece are plotted o the graph below. Which of the followig first order differece equatios could describe the sequece? a t + 1 = t + 1 with t 1 = 1 t b t + 1 = t + with t 1 = 1 10 c t + 1 = t with t 1 = 1 8 d t + 1 = t + 1 with t 1 = 6 e t + 1 = t + with t 1 = ebookplus Tutorial it-056 Worked example Term umber Thik 1 Elimiate the optios systematically. Examie the first term give by the graph to decide if it is t 1 = 1 or t 1 =. Observe ay patter betwee each successive poit o the graph. 3 Optio b gives both the correct patter ad first term. WriTe The coordiates of the first poit o the graph are (1, 1). The first term is t 1 = 1. Elimiate optios d ad e. There is a costat differece of + or t + 1 = t +. The aswer is b. 88 maths Quest 1 Further mathematics for the Casio Classpad

31 REMEMBER 1. A straight lie or liear patter is a arithmetic sequece give by first order differece equatios of the form: t + 1 = t + b. A o-liear (expoetial) patter is geerated by either of the followig: (a) a first order differece equatio of the form: t + 1 = a t (a geometric progressio) or (b) t + 1 = a t + b (a combiatio of a geometric ad a arithmetic sequece). Differece equatios 89

32 Exercise 6f Iterpretatio of the graph of first order differece equatios 1 WE 17 For each of the followig graphs, write a first order differece equatio that defies the sequece plotted o the graph. a t b t c t Term umber Term umber Term umber d e f t t t WE 18 For each of the followig graphs, write a first order differece equatio that defies the sequece plotted o the graph. a t b t c t Term umber Term umber Term umber Term umber Term umber d t e t f WE 19 MC The first five terms of a sequece are plotted o the graph at right. Which of the followig first order differece equatios could describe the sequece? a t + 1 = t + 1 t 1 = 1 b t + 1 = t t 1 = 1 c t + 1 = t + 1 t 1 = d t + 1 = t - 1 t 1 = e t + 1 = t + 1 t 1 = Term umber t Term umber Term umber t Term umber Term umber 90 Maths Quest 1 Further Mathematics for the Casio ClassPad

33 MC The first five terms of a sequece are plotted o the graph at right. Which of the followig first order differece equatios could describe the sequece? a t + 1 = t - 8 t 1 = 8 b t + 1 = t + 8 t 1 = 8 c t + 1 = t - 8 t 1 = 5 d t + 1 = t + 8 t 1 = 5 e t + 1 = 8t t 1 = 5 5 MC The first five terms of a sequece are plotted o the graph at right. Which of the followig first order differece equatios could describe the sequece? a t + 1 = - 0.5t t 1 = b t + 1 = 0.5t t 1 = 0.5 c t + 1 = t - 5 t 1 = 50 d t + 1 = - 0.5t t 1 = 50 e t + 1 = 0.5t t 1 = 50 6 MC The first five terms of a sequece are plotted o the graph at right. Which of the followig first order differece equatios could describe the sequece? a t = t t 1 = b t = 3t t 1 = c t = t t 1 = d t = 5t t 1 = e t = 6t t 1 = 7 MC The first five terms of a sequece are plotted o the graph at right. Which of the followig first order differece equatios could describe the sequece? a t = 0.5t - 1 t 1 = 0.5 b t = t t 1 = 8 c t = 8t - 1 t 1 = 0.5 d t = 0.5t - 1 t 1 = 8 e t = - 0.5t - 1 t 1 = 8 8 MC The first five terms of a sequece are plotted o the graph at right. Which of the followig first order differece equatios could describe the sequece? a t + 1 = 5t t 1 = 1 b t + 1 = t + 5 t 1 = 1 c t + 1 = 5 t 1 = 5 d t + 1 = 3t - 10 t 1 = 5 e t + 1 = 5t + 5 t 1 = 5 t t 50 0 t t t Term umber Term umber Term umber Term umber Term umber Differece equatios 91

34 6g 9 Graphs of the first five terms of first order differece equatios are show below together with the first order differece equatios. Match the graph with the first order differece equatio by writig the letter correspodig to the graph together with the umber correspodig to the first order differece equatio. a t b t c t d g Term umber t t Term umber Term umber i t + 1 = t + 1 t 1 = 8 ii t + 1 = t t 1 = 3 iii t + 1 = t - t 1 = 1 iv t + 1 = t t 1 = 1 v t + 1 = t t 1 = 1 vi t + 1 = - t t 1 = 5 vii t + 1 = t - 3 t 1 = viii t + 1 = t - 1 t 1 = 1 ix t + 1 = t - 5 t 1 = 6 e h Term umber t Term umber t Term umber f i Term umber t t Term umber Term umber Fiboacci sequeces as secod order differece equatios Fiboacci umbers Most of us have ever take the time to observe very carefully the umber or arragemets of petals ad seeds i flowers. If we were to do so, some very iterestig coclusios could be made. For each of the followig images, cout the umber of petals or spirals. 9 Maths Quest 1 Further Mathematics for the Casio ClassPad

35 Euphorbia has petals. Trillium has 3 petals. Lillies have 5 petals. Do you see a patter i the umbers so far? If you study the spirals i the photos of the cauliflower ad the pie coe, you will see that the spiral patter exists i the opposite directio also. What do you otice about the umber of spirals i the opposite directio i both of these photos? The umbers form part of a sequece of umbers kow as the Fiboacci umbers. The sequece of Fiboacci umbers is 1, 1,, 3, 5, 8, 13, 1, 3, 55, 89,... Each ew term i this sequece is formed by addig the two previous terms; the startig umbers are 1 ad 1. Ay sequece i which each ew term is the result of addig the previous two terms is kow as a Fiboacci sequece. For example, 5, 7, 1, 19, 31, 50,... is a Fiboacci sequece as each ew term is the sum of the previous two terms. A cauliflower has 8 spirals. Pie coes have 13 spirals. differece equatios 93

36 ebookplus Digital doc Ivestigatio Fiboacci patters Leoardo di Pisao, also kow as Fiboacci (which traslates as so of Boacci), first oticed the sequece of Fiboacci umbers i 10, whe he was asked by his kig to ivestigate a problem about how fast rabbits ca breed. The sequece of Fiboacci umbers ca be defied as a secod order differece equatio as follows: F + = F + F + 1 F 1 = 1 ad F = 1 As with first order differece equatios, this equatio cosists of two parts: F + = F + F + 1 describes the patter i the sequece (each ew term is formed by addig the two previous terms) F 1 = 1 ad F = 1 are the first two terms of the sequece. lucas umbers Aother useful Fiboacci sequece is oe called the Lucas umbers, amed after ieteethcetury mathematicia Edouard Lucas. The sequece of Lucas umbers starts with the umbers ad 1. The first 10 umbers of this sequece are, 1, 3,, 7, 11, 18, 9, 7 ad 76. Cotiue this sequece ad ivestigate the ratio of the terms L + 1. Compare this with the L ratio of the terms F + 1 from the sequece of Fiboacci umbers. What do you otice? F The sequece of Lucas umbers ca be also defied as a secod order differece equatio. L + = L + L + 1 L 1 = ad L = 1 As stated earlier, ay sequece i which each ew term is the result of addig the previous two terms, give ay two startig values, is kow as a Fiboacci sequece. The secod order differece equatio for a Fiboacci sequece is set out i the same way as defied earlier for the sequece of Fiboacci umbers ad Lucas umbers. The otatio ca be either of the followig: f + = f + f + 1 give f 1 ad f or t + = t + t + 1 give t 1 ad t Worked example 0 For the Fiboacci sequece give by the secod order differece equatio a state the first six terms of the sequece f + = f + f + 1 f 1 = ad f = 5, b use a CAS calculator to: i graphically display the first 6 terms ii fid the 1st term of the sequece; that is, f 1. Thik WriTe/diSplaY a 1 The first two terms are already defied. a f 1 = ad f = 5 Use the secod order differece equatio to geerate the remaiig required terms. f + = f + f + 1 f 3 = f 1 + f f = f + f 3 = + 5 = = 7 = 1 f 5 = f 3 + f f 6 = f + f 5 = = = 19 = 31 3 State the six terms. The first six terms of the sequece are, 5, 7, 1, 19 ad maths Quest 1 Further mathematics for the Casio Classpad

37 b i 1 Ope a Sequece scree. I b i the Recursive tab, tap ( ad complete the equatio as: a + = a + a + 1 a 1 = a = 5 Set the sequece table by tappig 8. Iput the values as show. To view the table, tap: OK _ To chage the widow settigs, tap 6. Set: xmi: 0 max: 7 ymi: 0 max: 35 The tap OK. To view the graph, tap: Graph G-Coect r (to view the full scree graph) ii 1 To chage the umber of ii terms, tap 8. Iput the values as show i the scree ad tap OK. Differece equatios 95

38 To view the values o a full scree, tap r. Write your aswer. The 1st term of the sequece is 187. As was the case with first order differece equatios, we ca use the secod order differece equatio for a Fiboacci sequece to fid the value of previous terms i a sequece, give that we have later umbers of the sequece. Worked Example 1 MC For part of a Fiboacci sequece give as, 9, 1, 3, 37, 60, 97, the first two terms could be give as: A t 1 =, t = 1 B t 1 = 1, t = C t 1 = - 1, t = - 3 D t 1 = - 1, t = 1 E t 1 =, t = 1 Thik 1 Write the differece equatio for the Fiboacci sequece. As we kow the later terms ad wish to fid the precedig terms, rearrage the differece equatio. 3 Substitute the kow values ito the equatio. This will produce the umber that precedes 9. Substitute 9 ad 5 ito the equatio to fid the umber that precedes 5. Cotiue with this process util the first two umbers match the two terms give i the questio. 5 The last two umbers calculated appear as optios; that is, t 1 = 1 ad t =. Write t + = t + t + 1 t = t + - t + 1 t = 1-9 = 5 t = 9-5 = t = 5 - = 1 The first two terms could be t 1 = 1 ad t =. Therefore, the aswer is B. A alterative method to solvig the questio i the above worked example could be a trial-aderror approach. Simply produce a sequece give the startig poits for each optio ad the see which oe results i the give sequece. 96 Maths Quest 1 Further Mathematics for the Casio ClassPad

39 Worked example Give t 1 = 3, t = 11 ad t 5 = 18 as three terms of a particular Fiboacci sequece, fid the value of t. Thik WriTe ebookplus Tutorial it-057 Worked example 1 Give the value of t ad t 5, work backwards by rearragig the differece equatio to fid t 3. t 5 = t 3 + t t 3 = t 5 - t = = 7 Use the values of t ad t 3 to fid t. t = t - t 3 = 11-7 = 3 State the aswer. The value of t is. remember The Fiboacci umbers are a uique sequece of umbers 1, 1,, 3, 5, 8, 13, 1, 3, 55, 89,... ad each ew term is formed by addig the two previous terms. Ay sequece whereby each ew term is the result of addig the previous two terms, give ay two startig values, is kow as a Fiboacci sequece. The Lucas umbers are a special group of umbers that follow a Fiboacci sequece ad have startig values of ad 1. The secod order differece equatio for a Fiboacci sequece is set out usig either of the followig otatios: f + = f + f + 1 give f 1 ad f or t + = t + t + 1 give t 1 ad t exercise 6G ebookplus Digital doc Spreadsheet 036 Fiboacci sequeces Fiboacci sequeces as secod order differece equatios 1 We0a For Fiboacci sequeces give by the secod order differece equatio f + = f + f + 1, give the first 10 terms whe the first two terms are defied as follows: a f 1 = 0 ad f = 1 b f 1 = ad f = - 1 c f 1 = 5 ad f = - 3 d f 1 = 3 ad f = - 1 State which of the above sequeces cotai the Fiboacci umbers. 3 For Fiboacci sequeces give by the secod order differece equatio f + = f + f + 1, list the first 10 terms whe the first two terms are defied as follows: a f 1 = 1 ad f = b f 1 = ad f = 0 c f 1 = ad f = 1 d f 1 = ad f = 5 We0b For the followig Fiboacci sequeces, use a CAS calculator to: i graphically display the first 6 terms ii fid the 1st term of the sequece; that is, f 1. a f + = f + f + 1 f 1 =, f = b f + = f + f + 1 f 1 = 1, f = 3 c f = f - + f - 1 f 1 = 3, f = 1 differece equatios 97

40 5 Use a appropriate method to graph the first eight terms of the Fiboacci sequeces with the followig startig terms: a f 1 = 6 ad f = 1 b f 1 = - 1 ad f = 8 c f 1 = - 1 ad f = - d f 1 = 3 ad f = - Commet o the shape of the graphs produced. 6 Write the first seve terms of the Fiboacci sequece i which: a the first two terms are ad 6 b the first two terms are 6 ad. Explai why the sequeces are differet eve though the same two values are used at the start. 7 MC For the differece equatio t = t - + t - 1, where t 1 = ad t = 3, the first five terms of the sequece are: a 3,, 7, 11, 18 b, 3,, 3, c, 3, 7, 10, 17 d, 3, 1, 36, 3 e 1, 7, 8, 15, 3 8 MC For the Fiboacci sequece with a differece equatio f = f - + f - 1, where f 1 = 1 ad f = 7, the value of f 7 is: a 61 b 51 c 38 d 3 e MC For the sequeces show below, which oe is a Fiboacci sequece? a 1, 3, 5, 7, 9, 11,... b 1,,, 8, 16, 3,... c 1, 3,, 7, 9, 11,... d 3, 3, 6, 9, 15,,... e 1,,, 5, 9, 18, WE1 MC For part of a Fiboacci sequece give as..., - 9, - 1, - 3, - 37, - 60, - 97, the first two terms could be give as: a t 1 = 1, t = - b t 1 = - 3, t = - 1 c t 1 = - 1, t = - 3 d t 1 = - 1, t = 1 e t 1 =, t = 1 11 MC For the sequece of Fiboacci umbers show i the graph, t Term umber the secod order differece equatio is: 6 0 a t = t - + t - 1, where t 1 = - 3 ad t = - 1 b t + = t + t + 1, where t 1 = - 1 ad t = - 3 c t + = t + t + 1, where t 1 = - 3 ad t = - 1 d t - = t + t + 1, where t 1 = - 1 ad t = e t = t - 1 -, where t 1 = - 1 ad t = Maths Quest 1 Further Mathematics for the Casio ClassPad

41 1 Usig a microscope to study the spread of a certai bacteria i a agar dish, a medical scietist observes the followig umber of coloies at the ed of each miute. After 1 miute After miutes After 3 miutes After miutes After 5 miutes Assumig the umber couted cotiues to follow this Fiboacci sequece, state the umber of bacteria (to the earest millio) expected after 30 miutes. 13 For each of the followig Fiboacci sequeces, determie the two startig terms, give that they both must be the smallest possible o-egative umbers. a..., 13,, 35, 57, 9 b..., 1, 3, 37, 60, 97 c..., 8, 15, 3, 38, 61 d..., 16, 5, 1, 66, WE Give the followig values as three terms of a particular Fiboacci sequece, fid the value of the required term. a t 1 =, t = 16 ad t 5 = 6; t =? Exam tip May studets fid the reverse b t = - 1, t = - 9 ad t 5 = - 17; t 1 =? applicatio of a differece equatio difficult; that is, they fid formulatig ad solvig a liear equatio c t = 3, t 5 = 7 ad t 6 = 1; t 1 =? demadig. d t =, t 6 = 57 ad t 7 = 9; t =? [Assessmet report 007] 15 Geerate a sequece of eight umbers usig the followig secod order differece equatios: a t + = t + t + 1 t 1 = 3, t = b t + = t + t + 1 t 1 = 1, t = 1 c t + = t + t + 1 t 1 = 1, t = d t = 3t - + t - 1 t 1 =, t = Differece equatios 99

42 Summary Geeratig the terms of a sequece defied by a first order differece equatio A first order differece equatio defies a relatioship betwee two successive terms of a sequece, for example, betwee: t, the previous term, ad t + 1, the ext term, or t - 1, the previous term, ad t, the ext term. A first order differece equatio has two mai parts: t + 1 = t + b (where b is a costat) describes the patter i the sequece t 1 = 1 is the first or a startig term i the sequece First order differece equatios ca be expressed as follows: t + 1 = t + 3 t 0 = 1 is read as the ext term is twice the previous term plus 3, startig at 1 or t t = t 1 = 1 is read as the differece betwee two cosecutive terms is, startig at 1. Startig term A startig term is eeded to fully defie a sequece. The same patter but differet startig poits gives differet sets of umbers. t + 1 = t +, t 1 = 3 gives 3, 5, 7, 9, 11,... t + 1 = t +, t 1 = gives,, 6, 8, 10,... t0 is used as the first term for situatios that are depedet o time. t1 is used as the first term for situatios that are ordial, such as placigs (first place, t 1, secod place, t, third place, t 3,...) or prizes (first prize, secod prize,...). The relatioship betwee arithmetic sequeces ad first order differece equatios Proumeral covetios Term Arithmetic ad geometric sequece covetio First order differece equatio covetio First term a or t 1 t 0 or t 1 Commo differece d b Commo ratio r a The commo differece, b = t - t 1 = t 3 - t = t - t 3 =... A arithmetic sequece with a commo differece of b may be defied by a first order differece equatio of the form: t + 1 = t + b (or t t = b) where b is the commo differece ad for b > 0 it is a icreasig sequece b < 0 it is a decreasig sequece. The relatioship betwee geometric sequeces ad first order differece equatios The geometric commo ratio, r, is the proumeral a i first order differece equatios. The commo ratio, r = a = t t3 t = = =... t1 t t3 300 Maths Quest 1 Further Mathematics for the Casio ClassPad

43 A geometric sequece with a commo ratio of a may be defied by a first order differece equatio of the form: t + 1 = at where a is the commo ratio a > 1 is a icreasig sequece 0 < a < 1 is a decreasig sequece a < 0 is a sequece alteratig betwee positive ad egative values. Settig up first order differece equatios: Type 1 Arithmetic sequece The ext term is the previous term plus a fixed amout or a fixed percetage of a iitial value. t + 1 = t + b where b = the commo differece = fixed amout or % of the first term, t 0 or t 1. Settig up first order differece equatios: Type Geometric sequece The ext term is the previous term plus a percetage of the previous term s value. t + 1 = t + % of t t + 1 = a t where a is the commo ratio. Settig up first order differece equatios: Type 3 Combiatio of arithmetic ad geometric sequece The ext term is the previous term plus a percetage of the previous term s value plus a fixed amout or a fixed percetage of a iitial value. t + 1 = t + % t + b t + 1 = a t + b where a = the commo ratio ad b = the commo differece. Iterpretatio of the graph of first order differece equatios A straight lie or liear patter is a arithmetic sequece give by first order differece equatios of the form t + 1 = t + b. A o-liear (expoetial) patter is geerated by either of the followig: (a) a first order differece equatio of the form: t + 1 = a t (a geometric progressio) Differece equatios 301

44 or (b) t + 1 = a t + b (a combiatio of a geometric ad a arithmetic sequece). Fiboacci sequeces as secod order differece equatios The Fiboacci umbers are a uique sequece of umbers (1, 1,, 3, 5, 8, 13, 1, 3, 55, 89,...). Each ew term is formed by addig the two previous terms. Ay sequece i which each ew term is the result of addig the previous two terms, give ay two startig values, is kow as a Fiboacci sequece. The Lucas umbers are a special group of umbers that follow a Fiboacci sequece ad have startig values of ad 1. The secod order differece equatio for a Fiboacci sequece is set out usig the followig otatio: f + = f + f + 1 give f 1 ad f or t + = t + t + 1 give t 1 ad t 30 Maths Quest 1 Further Mathematics for the Casio ClassPad

45 chapter review Multiple choice 1 Which oe of the followig equatios is ot a first order differece equatio? a t = - 1t - 1 b f + 1 = f - c p = p - 1 d t + 1 = 1 - e f = 10f - 1 Which of the sequeces below is geerated by the first order differece equatio t + 1 = t -, t 1 = - 6? a -, - 10, - 16, -, - 8,... b - 6, - 10, - 1, - 18, -, - 8,... c - 6, -, - 6, -, - 6,... d - 6, -,, 6, 10,... e - 6,, - 10,, - 1,... 3 A sequece is defied by the first order differece equatio: t + 1 = 3t - = 1,, 3,... The third term, (that is, t 3 ) of the sequece is - 7. The first term of the sequece is: a b c - 5 d - 1 e 5 3 The first order differece equatio that defies a arithmetic sequece is: a t + 1 = t t 1 = 1 b t + 1 = - t t 1 = 1 c t + 1 = t - t 1 = 1 d t + 1 = t + t 1 = 1 e t + 1 = - t + 3 t 1 = 1 5 The sequece,, 0, -, -,... ca be defied by the first order differece equatio: a t + 1 = t t 1 = b t + 1 = - t t 1 = c t + 1 = t - t 1 = d t + 1 = t + t 1 = e t + 1 = t + t 1 = 6 The first order differece equatio that defies a geometric sequece is: a t + 1 = t + t 1 = 1 b t + 1 = t - t 1 = 1 c t + 1 = t t 1 = 1 d t + 1 = - t + t 1 = 1 e t + 1 = t - t 1 = 1 7 The sequece - 3, 1, - 8, 19, - 768,... ca be defied by the first order differece equatio: a t + 1 = - t t 1 = - 3 b t + 1 = - t t 1 = 3 c t + 1 = - 3t t 1 = - d t + 1 = - 3t t 1 = e t + 1 = t t 1 = 3 8 A library adds 300 ew books to its collectio each year. The collectio bega with 000 books ad it is claimed that o book has ever bee removed. A first order differece equatio that reflects this situatio is: a B = B B 0 = 300 b B = B B 0 = 000 c B = 1.03B B 0 = 300 d B = 1.03B B 0 = 000 e B = 1.0B B 0 = George deposits $80 durig the secod week of each moth ito a accout that ears compoud iterest of 6% per aum compoudig mothly ad calculated at the ed of the moth. The first order differece equatio that would describe this situatio is: a A = 1.005A A 0 = 0 b A = 1.005A A 1 = 0 c A = 1.06A A 0 = 0 d A = 1.06A A 1 = 0 e A = 1.08A A 0 = 0 10 The first five terms of a sequece are plotted o the graph at right. The first order differece equatio that could describe the sequece is: a t + 1 = t - 0 t 1 = 7 b t + 1 = t + 0 t 1 = 7 c t + 1 = t - 0 t 1 = 0 d t + 1 = t - 7 t 1 = 0 e t + 1 = t + 7 t 1 = 0 11 The first five terms of a sequece are plotted o the graph at right. The first order differece equatio that could describe the sequece is: a t + 1 = t + t 1 = 3 b t + 1 = t + 1 t 1 = 3 c t + 1 = t - 3 t 1 = d t + 1 = t + 3 t 1 = e t + 1 = 3t t 1 = t t Term umber Term umber Differece equatios 303

46 1 For the Fiboacci sequece with a secod order differece equatio give as f + = f + f + 1, where f 1 = 3 ad f = 6, the value of f 8 is: a 165 c 63 e 13 b 10 d 1 13 For the sequeces show below, which oe is a Fiboacci sequece? a 1, 3, 5, 7, 9, 11,... b 1,,, 8, 16, 3,... c 1, 3,, 7, 11, 18, 9,... d 1, 3, 7, 15, 31, 63,... e 1, 1,,, 7, 13,... 1 For part of a Fiboacci sequece give as..., 1, 3, 37, 60, 97, the first two terms could be give as: a t 1 =, t = 6 b t 1 = 1, t = 7 c t 1 =, t = 3 d t 1 = 1, t = e t 1 = 0, t = 15 Paula starts a stamp collectio. She decides to buy a umber of ew stamps every week. The umber of stamps bought i the th week, t, is defied by the differece equatio t = t t - where t 1 = 1 ad t =. The total umber of stamps i her collectio after five weeks is: a 8 b 1 c 15 d 19 e [VCAA 006] 16 The rule for a differece equatio is P = P If P 6 = 1000, the P is equal to: a 100 b 00 c 600 d 1800 e 300 [VCAA 005] Short aswer 1 Write the first five terms of each of the sequeces defied below. a t = t t 1 = - 1 b t + 1 = 3 + 5t t 1 = 0 Express the sequece defied by t = -, = 1,, 3,... as a differece equatio. 3 Express the sequece defied by t = , = 1,, 3,... as a differece equatio. Show there is a commo ratio for the differece equatio t + 1 = 3t, where t 1 = 5. Exam tip Two calculatios are eeded to establish the commo ratio. For full marks studets had to show two quotiets. Some oly showed the calculatios ad did ot state the results. Others showed the results but did ot idicate their sigificace. [Assessmet report 007] 5 A club loses % of its membership each year but adds 0 ew members each year. The iitial membership of the club was 300. Write a differece equatio to describe this situatio, statig clearly the terms you use. 6 O a graph, plot the first five terms of the sequece described by the differece equatio: t + 1 = t - 1 t 1 = 7 The cost i dollars, C, to complete a housepaitig job o the th day is give by the differece equatio C + 1 = 0.5C where C = 300. a How much will it cost o day? b Show that it is either a arithmetic or geometric sequece. c How much will it cost o day 1? 8 The first four terms of t a sequece are plotted 0 o the graph at right. 30 Write the differece 0 equatio that defies 10 this sequece Term umber 9 Write the first five terms for each of the sequeces defied below: a t + = t + t + 1 t 1 = -, t = - 3 b f + = f + f + 1 f 1 = 0, f = c f = f - + f - 1 f 1 =, f = 6 10 Fid the sum of the first five terms of the sequece F + = F + F + 1 where F 1 = - 3 ad F = 6. Exam tip These types of questio require careful readig. This is a applicatio of a Fiboacci sequece which requires studets to determie the first five terms i the sequece ad the add them. May studets icorrectly gave the value of the fifth term oly, ot the sum of the five terms. [Assessmet report 006] 30 Maths Quest 1 Further Mathematics for the Casio ClassPad

47 Exteded respose Task 1 1 A bad has bee advised that, to tour successfully, the umber of gigs played i the th moth for the first five moths would eed to fit the differece equatio t = + 8, = 1,, 3... a Express this as a differece equatio. b Plot the 5 terms of the sequece. The occurrece of cymbal crashes i the bad s most popular rock ballad follows the geometric sequece t + 1 = t, t 1 = 3, where t is the bar umber of the th crash. The occurrece of timpai rolls i the same sog follows aother sequece, t + 1 = t + 10, t 1 =. At what bar umber does a timpai roll ad cymbal crash coicide for the first time? 3 As a future side-project, the lead guitarist of the bad wats to ope her ow record store. To fud the project, she sets up a savigs accout that ears 9% iterest p.a., compouded mothly ad calculated at the ed of the moth. She opes the accout with $500 ad deposits $100 at the start of each moth. Represet this iformatio with a differece equatio. The bad uses a exotic flower as the cover art for their ew record. The flower has a multi-tiered bud, where the umber of seeds per tier is give by the Fiboacci umbers 1, 1,, 3, 5, 8... a Represet this iformatio as a differece equatio. b Fid the umber of seeds i the 10th tier. c Which tier has 987 seeds? 5 The bad s maager otices that the sales of their ew record follow a Fiboacci sequece expressed by the differece equatio f = f f -, where f 1 = ad f =. Each term of the sequece is measured i thousads of uits shifted per moth. a List the first 8 terms of the sequece. b Determie the value of f 10. Task 1 Two brothers set up a small workshop to produce surfboards. The umber of surfboards they produce each moth follows a arithmetic sequece ad is give below. Moth umber Number of surfboards produced a Write the commo differece of the arithmetic sequece. b Write a differece equatio that defies this sequece, where B represets the umber of surfboards produced i the th moth. The brothers had bee advised that to be successful, the umber of surfboards produced i the th moth would eed to be reflected by B = - 1. a Express this as a differece equatio. b Plot the first five terms of the sequece. c I which moth would the umber of surfboards produced i a moth accordig to this advice first exceed the umber produced i a moth as described i questio 1? 3 I order to expad the busiess later, the brothers set up a savigs accout that ears 6% iterest per aum compouded mothly ad calculated at the ed of the moth. The brothers ope the accout with $1000 ad deposit $500 durig each moth. a Write a differece equatio to represet this iformatio, where A is the amout i the accout at the ed of the th moth. b Fid the amout i the accout after 5 years. Differece equatios 305

48 Task 3 Pythagorea triads are three itegers that satisfy Pythagoras theorem. These triads, such as 3, ad 5 or 5, 1 ad 13, ca be formed from a Fiboacci sequece as show below. Take ay four cosecutive terms of a Fiboacci sequece. To obtai the first umber of the Pythagorea triad, multiply the two middle terms ad double this aswer. To obtai the secod umber of the triad, multiply the two outer terms (from the four cosecutive terms). To obtai the third umber of the triad, sum the squares of the two middle terms (from the four cosecutive terms). 1 Cosider the small Fiboacci sequece 1,, 3 ad 5. State the values of t 1 ad t ad represet the sequece as a secod order differece equatio. Calculate the Pythagorea triad formed by the sequece 1,, 3 ad 5. 3 The terms t = 7 ad t 5 = 11 form part of a Fiboacci sequece. Fid the value of t 1, t ad t 3. Use t 1, t, t 3 ad t from questio 3 to fid the Pythagorea triad formed by the sequece of these four terms. 5 Calculate the Pythagorea triad formed by the sequece, 1, 3 ad. 6 What makes the sequece from questio 6 a Fiboacci sequece? Task 1 Maria decides to improve her fitess level by cyclig each day. The time i miutes, M, that Maria cycles o the th day is modelled by the differece equatio: M + 1 = 0.75M + 8 where M = 0. a For how may miutes will Maria cycle o day? b Show that the time Maria cycles each day does ot follow a arithmetic or a geometric sequece. c For how may miutes will Maria cycle o day 1? Rupert decides to iclude both swimmig ad ruig i his exercise pla. O day 1, Rupert swims 100 m ad rus 500 m. Each day he will icrease the distace he swims ad the distace he rus. His swimmig distace will icrease by 50 m each day. His ruig distace will icrease by % of the distace he ra o the previous day. O which day will the distace Rupert swims first be greater tha the distace he rus? [ Vcaa 007] ebookplus Digital doc Test Yourself 306 maths Quest 1 Further mathematics for the Casio Classpad

49 ebookplus activities chapter opeer Digital doc 10 Quick Questios: Warm up with a quick quiz o differece equatios. (page 59) 6c The relatioship betwee geometric sequeces ad first order differece equatios Digital doc WorkSHEET 6.1: Write arithmetic ad geometric sequeces i the form of differece equatios. (page 70) Tutorial We7 it-05: Watch a tutorial o recogisig the differece betwee geometric ad arithmetic sequeces give a differece equatio. (page 68) 6d Settig up first order differece equatios to represet practical situatios Digital doc SkillSHEET 6.1: Chagig a percetage to a decimal (page 77) Tutorials We 11 it-053: Lear how to use a differece equatio to model a simple iterest applicatio. (page 7) We 1 it-05: Lear how to use a differece equatio to model a bird populatio each year. (page 7) Iteractivity it-0187 Settig up first order differece equatios: Cosolidate your uderstadig of differece equatios. (page 70) 6e Graphical represetatio of a sequece defied by a first order differece equatio Digital doc WorkSHEET 6.: Recogise graphical represetatio of sequeces, solve worded problems ad fid terms give a differece equatio. (page 86) Tutorial We 1 it-055: Watch a tutorial o usig a CAS calculator to represet a sequece graphically. (page 80) 6F Iterpretatio of the graph of first order differece equatios Tutorial We 19 it-056: Watch a tutorial o fidig a first order differece equatio usig a graph. (page 88) 6G Fiboacci sequeces as secod order differece equatios Digital docs Ivestigatio: Fiboacci patters (page 9) Spreadsheet 036: Ivestigate terms i a Fiboacci sequece. (page 97) Tutorial We it-057: Watch a tutorial o fidig the value of a specific term give the differece equatio. (page 97) chapter review Digital doc Test Yourself: Take the ed-of-chapter test to test your progress. (page 306) To access ebookplus activities, log o to differece equatios 307