6Sequences. 6.1 Kick off with CAS 6.2 Arithmetic sequences 6.3 Geometric sequences 6.4 Recurrence relations 6.5 Review

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1 6Sequeces 6.1 Kick off with CAS 6.2 Arithmetic sequeces 6.3 Geometric sequeces 6.4 Recurrece relatios 6.5 Review

2 6.1 Kick off with CAS Explorig the Fiboacci sequece with CAS The Fiboacci sequece is a sequece of umbers that starts with 1 ad 1, after which every subsequet umber is foud by addig the two previous umbers. Thus the sequece is: 1, 1, 2, 3, 5, 8, 13, 21, 34, This sequece is frequetly foud i ature. For example, the umbers of petals of may flowers fall withi this sequece: a lily has 3 petals, a buttercup has 5 petals ad a daisy has 34 petals, just to ame a few. Withi the head of a suflower, seeds are produced at the cetre ad the migrate to the outside i spiral patters, with the umbers of seeds i the spirals beig umbers from the Fiboacci sequece. 1 Usig CAS ad a list ad spreadsheet applicatio, geerate the first 3 terms of the Fiboacci sequece. 2 If the first 3 umbers i the Fiboacci sequece are called t 1 = 1 (term 1), t 2 = 1 (term 2) ad t 3 = 2 (term 3), what is the value of t 2? 3 What is the smallest value of for which t > 1? 4 Calculate the ratios of cosecutive terms for the first 12 terms; that is, 1 1 = 1, 2 1 = 2, 3 2 = 1.5, 5 3 =?, =?, =?, =?, =?, =?, =?,?? =? 5 What do you otice about the value of the ratios as the terms icrease? Please refer to the Resources tab i the Prelims sectio of your ebookplus for a comprehesive step-by-step guide o how to use your CAS techology.

3 6.2 Uits 1 & 2 AOS 3 Topic 3 Cocept 1 Sequeces Cocept summary Practice questios Arithmetic sequeces Defiig mathematical sequeces A sequece is a related set of objects or evets that follow each other i a particular order. Sequeces ca be foud i everyday life, with some examples beig: the opeig share price of a particular stock each day the daily miimum temperature readigs i a particular city the lowest petrol prices each day the populatio of humas couted each year. Whe data is collected i the order that the evets occur, patters ofte emerge. Some patters ca be complicated, whereas others are easy to defie. I mathematics, sequeces are always ordered, ad the liks betwee differet terms of sequeces ca be idetified ad expressed usig mathematical equatios. You may already be familiar with some mathematical sequeces, such as the multiples of whole umbers or the square umbers. Multiples of 3: 3, 6, 9, 12, Multiples of 5: 5, 1, 15, 2, Square umbers: 1, 4, 9, 16, For each of these patters there is a lik betwee the umbers i the sequece (kow as terms) ad their positio i the sequece (kow as the term umber). The laguage of mathematical sequeces I geeral, mathematical sequeces ca be displayed as: t 1, t 2, t 3, t 4, t 5,..., t where t 1 is the first term, t 2 is the secod term, ad so o. The first term of a mathematical sequece ca also be referred to as a. The th term is referred to as t (so t 1 = a), ad represets the ordered positio of the term i the sequece, for example 1st, 2d, 3rd, Sequeces expressed as fuctios If we cosider the term umbers i a sequece as the iputs of a fuctio, the the term values of that sequece are the outputs of that fuctio. INPUT OUTPUT Term umber Fuctio Term value If we are able to defie a sequece as a fuctio, the we ca iput term umbers ito that fuctio to determie ay term value i the sequece. 212 Maths Quest 11 GENERAL MATHEMATICS VCE Uits 1 ad 2

4 WORKeD example 1 Determie the first five terms of the sequece t = THINK WRITE 1 Substitute = 1 ito the fuctio. t 1 = = 5 2 Substitute = 2 ito the fuctio. t 2 = = 7 3 Substitute = 3 ito the fuctio. t 3 = = 9 4 Substitute = 4 ito the fuctio. t 4 = = 11 5 Substitute = 5 ito the fuctio. t 5 = = 13 6 State the aswer The first five terms of the sequece are 5, 7, 9, 11 ad 13. Note: You ca see that the terms of the sequece icrease by the coefficiet of (i.e. the umber is multiplied by). Uits 1 & 2 AOS 3 Topic 3 Cocept 3 Arithmetic sequeces Cocept summary Practice questios arithmetic sequeces A arithmetic sequece is a sequece i which the differece betwee ay two successive terms i the sequece is the same. I a arithmetic sequece, the ext term i the sequece ca be foud by addig or subtractig a fixed value. First cosider the sequece 5, 9, 13, 17, 21. This is a arithmetic sequece, as each term is obtaied by addig 4 (a fixed value) to the precedig term. Now cosider the sequece 1, 3, 6, 1, 15. This is ot a arithmetic sequece, as each term does ot icrease by the same costat value. The commo differece The differece betwee two cosecutive terms i a arithmetic sequece is kow as the commo differece. If the commo differece is positive, the sequece is icreasig. If the commo differece is egative, the sequece is decreasig. I a arithmetic sequece, the first term is referred to as a ad the commo differece is referred to as d. WORKeD example 2 Determie which of the followig sequeces are arithmetic sequeces, ad for those sequeces which are arithmetic, state the values of a ad d. a 2, 5, 8, 11, 14, b 4, 1, 6, 11, 16, c 3, 5, 9, 17, 33, Topic 6 SeQueNCeS 213

5 THINK a 1 Calculate the differece betwee cosecutive terms of the sequece. 2 If the differeces betwee cosecutive terms are costat, the the sequece is arithmetic. The first term of the sequece is a ad the commo differece is d. b 1 Calculate the differece betwee cosecutive terms of the sequece. 2 If the differeces betwee cosecutive terms are costat, the the sequece is arithmetic. The first term of the sequece is a ad the commo differece is d. c 1 Calculate the differece betwee cosecutive terms of the sequece. 2 If the differeces betwee cosecutive terms are costat, the the sequece is arithmetic. WRITE a t 2 t 1 = 5 2 = 3 t 3 t 2 = 8 5 = 3 t 4 t 3 = 11 8 = 3 t 5 t 4 = = 3 The commo differeces are costat, so the sequece is arithmetic. a = 2 ad d = 3 b t 2 t 1 = 1 4 = 5 t 3 t 2 = 6 1 = = 5 t 4 t 3 = 11 6 = = 5 t 5 t 4 = = = 5 The commo differeces are costat, so the sequece is arithmetic. a = 4 ad d = 5 c t 2 t 1 = 5 3 = 2 t 3 t 2 = 9 5 = 4 t 4 t 3 = 17 9 = 8 t 5 t 4 = = 16 The commo differeces are ot costat, so the sequece is ot arithmetic. Equatios represetig arithmetic sequeces If we wat to determie ay term of a arithmetic sequece, we eed to set up a equatio to represet the sequece. Ay arithmetic sequece ca be expressed by the equatio t = a + ( 1)d, where t is the th term, a is the first term ad d is the commo differece. Therefore, if we kow or ca determie the values of a ad d, we ca costruct the equatio for the sequece. 214 Maths Quest 11 GENERAL MATHEMATICS VCE Uits 1 ad 2

6 WORKeD example 3 Determie the equatios that represet the followig arithmetic sequeces. a 3, 6, 9, 12, 15, b 4, 33, 26, 19, 12, THINK WRITE a 1 Determie the values of a ad d. a a = 3 d = t 2 t 1 = 6 3 = 3 2 Substitute the values for a ad d ito the formula for arithmetic sequeces. t = a + ( 1)d = 3 + ( 1) 3 = 3 + 3( 1) = = 3 b 1 Determie the values of a ad d. b a = 4 d = t 2 t 1 = 33 4 = 7 2 Substitute the values for a ad d ito the formula for arithmetic sequeces. t = a + ( 1)d = 4 + ( 1) 7 = 4 7( 1) = = 47 7 Iteractivity Terms of a arithmetic sequece it-6261 Determiig future terms of a arithmetic sequece After a equatio has bee set up to represet a arithmetic sequece, we ca use this equatio to determie ay term i the sequece. Simply substitute the value of ito the equatio to determie the value of that term. Determiig other values of a arithmetic sequece We ca obtai the values a, d ad for a arithmetic sequece by trasposig the equatio. a = t ( 1)d d = t a 1 = t a d + 1 WORKeD example 4 a Fid the 15th term of the sequece 2, 8, 14, 2, 26, b Fid the first term of the arithmetic sequece i which t 22 = 18 ad d = 8. Topic 6 SeQueNCeS 215

7 c Fid the commo differece of the arithmetic sequece which has a first term of 12 ad a 11th term of 12. d A arithmetic sequece has a first term of 4 ad a commo differece of 12. Which term umber has a value of 196? THINK a 1 As it has a commo differece, this is a arithmetic sequece. State the kow values. 2 Substitute the kow values ito the equatio for a arithmetic sequece ad solve. WRITE a a = 2, d = 6, = 15 t = a + ( 1)d t 15 = 2 + (15 1)6 = = = 86 3 State the aswer The 15th term of the sequece is 86. b 1 State the kow values of the arithmetic sequece. 2 Substitute the kow values ito the equatio to determie the first term ad solve. b d = 8, = 22, t 22 = 18 a = t ( 1)d = 18 (22 1)( 8) = 18 (21)( 8) = = = State the aswer The first term of the sequece is c 1 State the kow values of the arithmetic sequece. 2 Substitute the kow values ito the equatio to determie the commo differece ad solve. c a = 12, = 11, t 11 = 12 d = t a = 11 1 = 9 1 = 9 3 State the aswer The commo differece is 9. d 1 State the kow values of the arithmetic sequece. 2 Substitute the kow values ito the equatio to determie the term umber ad solve. d a = 4, d = 12, t = 196 = t a + 1 d = = 14 3 State the aswer The 14th term i the sequece has a value of Maths Quest 11 GENERAL MATHEMATICS VCE Uits 1 ad 2

8 Graphical displays of sequeces Tables of values Whe we draw a graph of a mathematical sequece, it helps to first draw a table of values for the sequece. The top row of the table displays the term umber of the sequece, ad the bottom of the table displays the term value. Term umber Term value The data from the table of values ca the be used to idetify the poits to plot i the graph of the sequece. Drawig graphs of sequeces Whe we draw a graph of a umerical sequece, the term umber is the idepedet variable, so it appears o the x-axis of the graph. The term value is the depedet value, so it appears o the y-axis of the graph. Iteractivity Arithmetic sequeces it-6258 WORKeD example 5 Graphical displays of arithmetic sequeces Because there is a commo differece betwee the terms of a arithmetic sequece, the relatioship betwee the terms is a liear relatioship. This meas that whe we graph the terms of a arithmetic sequece, we ca joi the poits to form a straight lie. Whe we draw a graph of a arithmetic sequece, we ca exted the straight lie to determie values of terms i the sequece that have t yet bee determied. A arithmetic sequece is give by the equatio t = 7 + 2( 1). a Draw up a table of values showig the term umber ad term value for the first 5 terms of the sequece. b Plot the graph of the sequece. c Use your graph of the sequece to determie the 12th term of the sequece. THINK WRITE/dRaW a 1 Set up a table with the term umber i the top row ad the term value i the bottom row. 2 Substitute the first 5 values of ito the equatio to determie the missig values. a Term umber Term value t 1 = 7 + 2(1 1) = = 7 + = 7 t 2 = 7 + 2(2 1) = = = 9 t 3 = 7 + 2(3 1) = = = 11 Topic 6 SeQueNCeS 217

9 3 Complete the table with the calculated values. b 1 Use the table of values to idetify the poits to be plotted. 2 Plot the poits o the graph. t 4 = 7 + 2(4 1) = = = 13 t 5 = 7 + 2(5 1) = = = 15 Term umber Term value b The poits to be plotted are (1, 7), (2, 9), (3, 11), (4, 13) ad (5, 15). Term value 32 t Term umber c 1 Joi the poits with a straight lie ad exted the lie to cover future values of the sequece. c Term value 32 t Term umber 218 Maths Quest 11 GENERAL MATHEMATICS VCE Uits 1 ad 2

10 2 Read the required value from the graph (whe = 12). Term value 32 t Term umber 3 Write the aswer. The 12th term of the sequece is 29. Uits 1 & 2 AOS 3 Topic 3 Cocept 4 Modellig usig arithmetic sequeces Cocept summary Practice questios WORKeD example 6 usig arithmetic sequeces to model practical situatios If we have a practical situatio ivolvig liear growth or decay i discrete steps, this situatio ca be modelled by a arithmetic sequece. Simple iterest As covered i Topic 3, simple iterest is calculated o the origial amout of moey ivested. It is a fixed amout of iterest paid at equal itervals, ad as such it ca be modelled by a arithmetic sequece. Remember that simple iterest is calculated by usig the formula I = PrT, where I is 1 the amout of simple iterest, P is the priciple, r is the percetage rate ad T is the amout of periods. Jelea puts $1 ito a ivestmet that ears simple iterest at a rate of.5% per moth. a Set up a equatio that represets Jelea s situatio as a arithmetic sequece, where t is the amout i Jelea s accout after moths. b Use your equatio from part a to determie the amout i Jelea s accout at the ed of each of the first 6 moths. c Calculate the amout i Jelea s accout at the ed of 18 moths. Topic 6 SeQueNCeS 219

11 THINK a 1 Use the simple iterest formula to determie the amout of simple iterest Jelea ears i oe moth. 2 Calculate the amout i the accout after the first moth. 3 State the kow values i the arithmetic sequece equatio. 4 Substitute these values ito the arithmetic sequece equatio. b 1 Use the equatio from part a to fid the values of t 2, t 3, t 4, t 5 ad t 6. WRITE a I = PrT = 1 = 5 1 = 5 a = = 15 a = 15, d = 5 t = ( 1) b t 2 = (2 1) = = = 11 t 3 = (3 1) = = = 115 t 4 = (4 1) = = = 12 t 5 = (5 1) = = = 125 t 6 = (6 1) = = = 13 2 Write the aswer. The amouts i Jelea s accout at the ed of each of the first 6 moths are $15, $11, $115, $12, $125 ad $13. c 1 Use the equatio from part a to fid the values of t 18. c t 18 = (18 1) = = = 19 2 Write the aswer. After 18 moths Jelea has $19 i her accout. 22 Maths Quest 11 GENERAL MATHEMATICS VCE Uits 1 ad 2

12 Uits 1 & 2 AOS 3 Topic 3 Cocept 7 Depreciatio Cocept summary Practice questios Depreciatig assets May items, such as automobiles or electroic equipmet, decrease i value over time as a result of wear ad tear. At tax time idividuals ad compaies use depreciatio of their assets to offset expeses ad to reduce the amout of tax they have to pay. Uit cost depreciatio Uit cost depreciatio is a way of depreciatig a asset accordig to its use. For example, you ca depreciate the value of a car based o how may kilometres it has drive. The uit cost is the amout of depreciatio per uit of use, which would be 1 kilometre of use i the example of the car. Future value ad write-off value Whe depreciatig the values of assets, compaies will ofte eed to kow the future value of a item. This is the value of that item at that specific time. The write-off value or scrap value of a asset is the poit at which the asset is effectively worthless (i.e. has a value of $) due to depreciatio. WORKeD example 7 Loi purchases a ew car for $25 ad decides to depreciate it at a rate of $.2 per km. a Set up a equatio to determie the value of the car after km of use. b Use your equatio from part a to determie the future value of the car after it has 75 km o its clock. THINK a 1 Calculate the value of the car after 1 km of use. 2 State the kow values i the arithmetic sequece equatio. 3 Substitute these values ito the arithmetic sequece equatio. WRITE a a = 25.2 = a = , d =.2 t = a + ( 1)d = ( 1).2 = ( 1) b 1 Substitute = 75 ito the equatio b t = ( 1) determied i part a. t 75 = (75 1) = = = 23 2 Write the aswer. After 75 km the car will be worth $23. ExERCIsE 6.2 PRaCTIsE Arithmetic sequeces 1 WE1 Determie the first five terms of the sequece t = Determie the first five terms of the sequece t = 3 5. Topic 6 SeQueNCeS 221

13 3 WE2 Determie which of the followig sequeces are arithmetic sequeces, ad for those sequeces which are arithmetic, state the values of a ad d. a 23, 68, 113, 158, 23, c 1 2, 3 4, 1, 5 4, 3 2, 7 4,... b 3, 8, 23, 68, 23, 4 Fid the missig values i the followig arithmetic sequeces. a 13, 12, 37, f, 87, b 2.5, j, 8.9, 12.1, k, c p, q, r, 9 2, 25 4,... 5 WE3 Determie the equatios that represet the followig arithmetic sequeces. a 1, 3, 7, 11, 15, b 1.5, 2, 5.5, 8, 11.5 c 7 2, 11 2, 15 2, 19 2, 23 2,... 6 Determie the first five terms of the followig arithmetic sequeces. a t = 5 + 3( 1) b t = 1 7( 1) c t = ( 1) 3 7 WE4 a Fid the 2th term of the sequece 85, 72, 59, 46, 33, b Fid the first value of the arithmetic sequece i which t 7 = 5 ad d = a Fid the commo differece of the arithmetic sequece that has a first term of 32 ad a 8th term of 34. b A arithmetic sequece has a first term of 5 ad a commo differece of 4. Which term umber has a value of 85? c A arithmetic sequece has a first term of 4 ad a commo differece of 12. Which term umber has a value of 196? 9 WE5 A arithmetic sequece is give by the equatio t = 5 + 1( 1). a Draw up a table of values showig the term umber ad term value for the first 5 terms of the sequece. b Plot the graph of the sequece. c Use your graph of the sequece to determie the 9th term of the sequece. 1 A arithmetic sequece is defied by the equatio t = ( 1). a Draw up a table of values showig the term umber ad term value for the first 5 terms of the sequece. b Plot the graph of the sequece. c Use your graph of the sequece to determie the 13th term of the sequece. 11 WE6 Grigor puts $15 ito a ivestmet accout that ears simple iterest at a rate of 4.8% per year. a Set up a equatio that represets Grigor s situatio as a arithmetic sequece, where t is the amout i Grigor s accout after moths. b Use your equatio from part a to determie the amout i Grigor s accout after each of the first 6 moths. c Calculate the amout i Grigor s accout at the ed of 18 moths. 12 Justie sets up a equatio to model the amout of her moey i a simple iterest ivestmet accout after moths. Her equatio is t = ( 1), where t is the amout i Justie s accout after moths. a How much did Justie ivest i the accout? b What is the aual iterest rate of the ivestmet? 222 Maths Quest 11 GENERAL MATHEMATICS VCE Uits 1 ad 2

14 Cosolidate 13 WE7 Phillipe purchases a ew car for $24 ad decides to depreciate it at a rate of $.25 per km. a Set up a equatio to determie the value of the car after km of use. b Use your equatio from part a to determie the future value of the car after it has 12 km o its clock. 14 Dougie is i charge of the equipmet for his office. He decides to depreciate the value of a photocopier at the rate of x cets for every copies made. Dougie s equatio for the value of the photocopier after copies is t = ( 1) a How much did the photocopier cost? b What is the rate of depreciatio per copy made? 15 a Fid the 15th term of the arithmetic sequece 6, 13, 2, 27, 34, b Fid the 2th term of the arithmetic sequece 9, 23, 37, 51, 65, c Fid the 3th term of the arithmetic sequece 56, 48, 4, 32, 24, d Fid the 55th term of the arithmetic sequece 72 5, 551 4, 263 2, 51 4, 119 1, a Fid the first value of the arithmetic sequece which has a commo differece of 6 ad a 31st term of 94. b Fid the first value of the arithmetic sequece which has a commo differece of 2 ad a 4th term of c Fid the commo differece of a arithmetic sequece which has a first value of 564 ad a 51st term of 54. d Fid the commo differece of a arithmetic sequece which has a first value of 87 ad a 61st term of a A arithmetic sequece has a first value of 12 ad a commo differece of 16. Which term has a value of 712? b A arithmetic sequece has a first value of 32 ad a commo differece of 4. Which term has a value of 116? 18 Three cosecutive terms of a arithmetic sequece are x 5, x + 4 ad 2x 7. Fid the value of x. 19 The graph shows some poits of a arithmetic sequece. a What is the commo differece betwee cosecutive terms? b What is the value of the first term of the sequece? c What is the value of the 12th term of the sequece? Term value 16 t Term umber Topic 6 Sequeces 223

15 2 Sketch the graph of t = a + ( 1)d, where a = 15 ad d = 25, for the first 1 terms. 21 A employee starts a ew job with a $6 salary i the first year ad the promise of a pay rise of $25 a year. a How much will her salary be i their 6th year? b How log will it take for her salary to reach $85? 22 Nadia wats to ivest her moey ad decided to place $9 ito a credit uio accout earig simple iterest at a rate of 6% per year. a How much iterest will Nadia receive after oe year? b What is the total amout Nadia has i the credit uio after years? c For how log should Nadia keep her moey ivested if she wats a total of $154 8 retured? 23 Tom bought a car for $23, kowig it would depreciate i value by $21 per moth. a What is the value of the car after 18 moths? b By how much does the value of the car depreciate i 3 years? c How may moths will it take for the car to be valued at $62? 24 A cofectioary maufacturer itroduces a ew sweet ad produces 5 packets of the sweets i the first week. The stores sell them quickly, ad i the followig week there is demad for 3% more. I each subsequet week the icrease i productio is 3% of the origial productio. a How may packs are maufactured i the 2th week? b I which week will the cofectioary maufacturer produce 5 54 packs? Master 25 A caig machie was purchased for a total of $25 ad is expected to produce 5 cas before it is writte off. a By how much does the caig machie depreciate with each ca made? b If the caig machie were to make 4 2 cas each year, whe will the machie theoretically be writte off? c Whe will the machie have a book value of $89 2? 26 The local rugby club wats to icrease its membership. I the first year they had 5 members, ad so far they have maaged to icrease their membership by 12 members per year. a If the icrease i membership cotiues at the curret rate, how may members will they have i 15 years time? 224 Maths Quest 11 GENERAL MATHEMATICS VCE Uits 1 ad 2

16 Tickets for membership i the first year were $2, ad each year the price has rise by a costat amout, with memberships i the 6th year costig $32. b How much would the tickets cost i 15 years time? c What is the total membership icome i both the first ad 15th years? 6.3 Uits 1 & 2 AOS 3 Topic 3 Cocept 5 Geometric sequeces Cocept summary Practice questios Geometric sequeces Geometric sequeces A geometric sequece is a patter of umbers whose cosecutive terms icrease or decrease i the same ratio. First cosider the sequece 1, 3, 9, 27, 81, This is a geometric sequece, as each term is obtaied by multiplyig the precedig term by 3. Now cosider the sequece 1, 3, 6, 1, 15, This is ot a geometric sequece, as the cosecutive terms are ot icreasig i the same ratio. Commo ratios The ratio betwee two cosecutive terms i a geometric sequece is kow as the commo ratio. I a geometric sequece, the first term is referred to as a ad the commo ratio is referred to as r. WORKeD example 8 Determie which of the followig sequeces are geometric sequeces, ad for those sequeces which are geometric, state the values of a ad r. a 2, 4, 8, 16, 32, b 8, 4, 2, 1, 1 2,... c 3, 9, 27, 81, d 2, 4, 6, 8, 1, THINK a 1 Calculate the ratio t +1 betwee all cosecutive terms i the sequece. t 2 If the ratios betwee cosecutive terms are costat, the the sequece is geometric. The first term of the sequece is a ad the commo differece is r. WRITE a t 2 = 4 t 1 2 = 2 t 3 = 8 t 2 4 = 2 t 4 = 16 t 3 8 = 2 t 5 = 32 t 4 16 = 2 The ratios betwee cosecutive terms are all 2, so this is a geometric sequece. a = 2, r = 2 Topic 6 SeQueNCeS 225

17 b 1 Calculate the ratio t +1 betwee all cosecutive terms i the sequece. t b t 2 t 1 = 4 8 = 1 2 t 3 t 2 = 2 4 t 4 = 1 2 t 3 = 1 2 t 5 t 4 = = If the ratios betwee cosecutive terms are costat, the the sequece is geometric. The first term of the sequece is a ad the commo differece is r. The ratios betwee cosecutive terms are all 1 so this is a geometric sequece. 2 c 1 Calculate the ratio t +1 betwee all cosecutive terms i the sequece. t 2 If the ratios betwee cosecutive terms are costat, the the sequece is geometric. The first term of the sequece is a ad the commo differece is r. a = 8, r = 1 2 c t 2 = 9 t 1 3 = 3 t 3 = 27 t 2 9 = 3 t 4 = 81 t 3 27 = 3 The ratios betwee cosecutive terms are all 3, so this is a geometric sequece. a = 3, r = 3 d 1 Calculate the ratio t +1 betwee all cosecutive terms i the sequece. t d t 2 = 4 t 1 2 = 2 t 3 = 6 t 2 4 = 3 2 t 4 = 8 t 3 6 = 4 3 t 5 = 1 t 4 8 = Maths Quest 11 GENERAL MATHEMATICS VCE Uits 1 ad 2

18 2 If the ratios betwee cosecutive terms are costat, the the sequece is geometric. All of the ratios betwee cosecutive terms are differet, so this is ot a geometric sequece. equatios represetig geometric sequeces Ay geometric sequece ca be represeted by the equatio t = ar 1, where t is the th term, a is the first term ad r is the commo ratio. Therefore, if we kow or ca determie the values of a ad r for a geometric sequece, we ca costruct the equatio for the sequece. WORKeD example 9 Determie the equatios that represet the followig geometric sequeces. a 7, 28, 112, 448, 1792, b 8, 4, 2, 1, 1 2, THINK WRITE a 1 Determie the values of a ad r. a a = 7 r = t 2 t 1 = 28 7 = 4 2 Substitute the values for a ad r ito the formula for geometric sequeces. t = ar 1 = b 1 Determie the values of a ad r. b a = 8 r = t 2 t 1 = 4 8 = Substitute the values for a ad r ito the formula for geometric sequeces. t = ar 1 = Iteractivity Terms of a geometric sequece it-626 Determiig future terms of a geometric sequece After a equatio has bee set up to represet a geometric sequece, we ca use this equatio to determie ay term i the sequece. Simply substitute the value of ito the equatio to determie the value of that term. Determiig other values of a geometric sequece We ca obtai the values a ad r for a geometric sequece by trasposig the equatio. Topic 6 SeQueNCeS 227

19 a = r = t r 1 t a 1 1 Note: The value of ca also be determied, but this is beyod the scope of this course. WORKeD example 1 a Fid the 2th term of the geometric sequece with a = 5 ad r = 2. b A geometric sequece has a first term of 3 ad a 2th term of Fid the commo ratio betwee cosecutive terms of the sequece. c Fid the first term of a geometric series with a commo ratio of 2.5 ad a 5th term of THINK WRITE a 1 Idetify the kow values i the questio. a a = 5 r = 2 = 2 2 Substitute these values ito the geometric sequece formula ad solve to fid the missig value. t = ar 1 t 2 = = = Write the aswer. The 2th term of the sequece is b 1 Idetify the kow values i the questio. b t 2 = a = 4 = 2 2 Substitute these values ito the formula to calculate the commo ratio ad solve to fid the missig value. r = = t a = = Write the aswer. The commo ratio betwee cosecutive terms of the sequece is 2. c 1 Idetify the kow values i the questio. c t 5 = r = 2.5 = MaThS QueST 11 GeNeRaL MaTheMaTICS VCe uits 1 ad 2

20 2 Substitute these values ito the formula to calculate the first term ad solve to fid the missig value. a = t 1 r = = = = 3 3 Write the aswer. The first term of the sequece is 3. Iteractivity Geometric sequeces it-6259 Graphs of geometric sequeces The shape of the graph of a geometric sequece depeds o the value of r. Whe r > 1, the values of the terms icrease or decrease at a expoetial rate. Whe < r < 1, the values of the terms coverge towards. Whe 1 < r <, the values of the terms oscillate o either side of but coverge towards. Whe r < 1, the values of the terms oscillate o either side of ad move away from the startig value at a expoetial rate. r > 1 < r < 1 t t Term value Term value Term umber Term umber t 1 < r < t r < 1 Term value Term umber Term value Term umber Topic 6 Sequeces 229

21 WORKeD example 11 A geometric sequece is defied by the equatio t = a Draw up a table of values showig the term umber ad term value for the first 5 terms of the sequece. b Plot the graph of the sequece. THINK a 1 Set up a table with the term umber i the top row ad the term value i the bottom row. 2 Substitute the first 5 values of ito the equatio to determie the missig values. 3 Complete the table with the calculated values. b 1 Use the table of values to idetify the poits to be plotted. 2 Plot the poits o the graph. WRITE/dRaW a Term umber Term value t 1 = = 5 2 = 5 1 = 5 t 2 = = = 5 2 = 1 t 3 = = = 5 4 = 2 t 4 = = = 5 8 = 4 t 1 = = = 5 16 = 8 Term umber Term value b The poits to be plotted are (1, 5), (2, 1), (3, 2), (4, 4) ad (5, 8). Term value 9 t Term umber 23 MaThS QueST 11 GeNeRaL MaTheMaTICS VCe uits 1 ad 2

22 Uits 1 & 2 AOS 3 Topic 3 Cocept 6 Modellig usig geometric sequeces Cocept summary Practice questios WORKeD example 12 usig geometric sequeces to model practical situatios If we have a practical situatio ivolvig geometric growth or decay i discrete steps, this situatio ca be modelled by a geometric sequece. Compoud iterest As covered i Topic 3, compoud iterest is calculated o the sum of a ivestmet at the start of each compoudig period. The amout of iterest accrued varies throughout the life of the ivestmet ad ca be modelled by a geometric sequece. Remember that simple iterest is calculated by usig the formula A = P 1 + r, where A is 1 the total amout of the ivestmet, P is the priciple, r is the percetage rate ad is the umber of compoudig periods. Alexis puts $2 ito a ivestmet accout that ears compoud iterest at a rate of.5% per moth. a Set up a equatio that represets Alexis s situatio as a geometric sequece, where t is the amout i Alexis accout after moths. b Use your equatio from part a to determie the amout i Alexis s accout at the ed of each of the first 6 moths. c Calculate the amout i Alexis s accout at the ed of 15 moths. THINK a 1 Determie the amouts i the accout after each of the first two moths. 2 Calculate the commo ratio betwee cosecutive terms. WRITE a A = P 1 + r 1 = = = 21 A = P 1 + r 1 = = = 22.5 r = t 2 t 1 = = State the kow values i the geometric sequece equatio. a = 21, r = 1.5 Topic 6 SeQueNCeS 231

23 4 Substitute these values ito the geometric sequece equatio. t = b 1 Use the equatio from part a to fid the values of t 3, t 4, t 5 ad t 6. Roud all values correct to 2 decimal places. b t 3 = = = = t 4 = = = = t 5 = = = = t 6 = = = = Write the aswer. The amouts i Alexis accout at the ed of each of the first 6 moths are $21, $22.5, $23.15, $24.3, $25.5 ad $ c 1 Use the equatio from part a to fid the values of t 15, roudig your aswer correct to 2 decimal place. c t 15 = = = = Write the aswer. After 15 moths Alexis has $ i her accout. Note: The commo ratio i the geometric sequece equatio is equal to 1 + r (from the compoud iterest formula). 1 Reducig balace depreciatio Aother method of depreciatio is reducig balace depreciatio. Whe a item is depreciated by this method, rather tha the value of the item depreciatig by a fixed amout each year, it depreciates by a percetage of the previous future value of the item. Due to the ature of reducig balace depreciatio, we ca represet the sequece of the future values of a item that is beig depreciated by this method as a geometric sequece. 232 Maths Quest 11 GENERAL MATHEMATICS VCE Uits 1 ad 2

24 WORKeD example 13 A hot water system purchased for $125 is depreciated by the reducig balace method at a rate of 8% p.a. a Set up a equatio to determie the value of the hot water system after years of use. b Use your equatio from part a to determie the future value of the hot water system after 6 years of use (correct to the earest cet). THINK a 1 Calculate the commo ratio by idetifyig the value of the item i ay give year as a percetage of the value i the previous year. Covert the percetage to a ratio by dividig by 1. 2 Calculate the value of the hot water system after 1 year of use. 3 Substitute the values of a ad r ito the geometric sequece equatio. b 1 Substitute = 6 ito the equatio determied i part a. Give your aswer correct to 2 decimal places. WRITE a 1% 8% = 92% Each year the value of the item is 92% of the previous value. 92% = 92 1 =.92 r =.92 b a = = 115 t = t = t 6 = = = Write the aswer. After 6 years the book value of the hot water system is $ ExERCIsE 6.3 PRaCTIsE Geometric sequeces 1 WE8 Determie which of the followig sequeces are geometric sequeces, ad for those sequeces which are geometric, state the values of a ad r. a 3, 6, 12, 24, 48, b 1 2, 5 4, 25 8, ,... c 9, 6, 3,, 3, d 1 2, 1 5, 2 25, 4 125,... 2 Fid the missig values i the followig geometric sequeces. a 1, 6, c, 216, 1296 b 3, g, h, 24, 48 c p, q, s, 3, 15 3 WE9 Determie the equatios that represet the followig geometric sequeces. a 1, 5, 25, 125, 625, b 7, 3.5, 1.75,.875,.4375 c 5 6, 5 9, 1 27, 2 81, 4 243,... 4 Determie the first five terms of the followig arithmetic sequeces. 1 1 a t = b t = c t = Topic 6 SeQueNCeS 233

25 5 WE1 a Fid the 15th term of the geometric sequece with a = 4 ad r = 3. b A geometric sequece has a first term of 2 ad a 12th term of Fid the commo ratio betwee cosecutive terms of the sequece. c Fid the first term of a geometric series with a commo ratio of 1 2 ad a 6th term of a Fid the 11th term of the geometric sequece with a first value of 1.2 ad a commo ratio of 4. b A geometric sequece has a first term of 1.5 ad a 1th term of 768. Fid the commo ratio betwee cosecutive terms of the sequece. c Fid the first term of a geometric series with a commo ratio of.4 ad a 6th term of WE11 A geometric sequece is defied by the equatio t = a Draw a table of values showig the term umber ad term value for the first 5 terms of the sequece. b Plot the graph of the sequece. 8 A geometric sequece is defied by the equatio t = a Draw a table of values showig the term umber ad term value for the first 5 terms of the sequece. b Plot the graph of the sequece. 9 WE12 Hussei puts $25 ito a ivestmet that ears compoud iterest at a rate of.3% per moth. a Set up a equatio that represets Hussei s situatio as a geometric sequece, where t is the amout i Hussei s accout after moths. b Use your equatio from part a to determie the amout i Hussei s accout after each of the first 6 moths. c Calculate the amout i Hussei s accout at the ed of 15 moths. 1 Tim sets up a equatio to model the amout of his moey i a compoud iterest ivestmet accout after moths. His equatio is t = , where t is the amout i his accout after moths. a How much did Tim ivest i the accout? b What is the aual iterest rate of the ivestmet? 11 WE13 A refrigerator purchased for $147 is depreciated by the reducig balace method at a rate of 7% p.a. a Set up a equatio to determie the value of the refrigerator after years of use. b Use your equatio from part a to determie the future value of the refrigerator after 8 years of use. 12 Ivy buys a ew ove ad decides to depreciate the value of the ove by the reducig balace method. Ivy s equatio for the value of the ove after years is t = a How much did the ove cost? b What is the aual rate of depreciatio for the ove? 234 Maths Quest 11 GENERAL MATHEMATICS VCE Uits 1 ad 2

26 Cosolidate 13 Which of the followig are geometric sequeces? Where applicable state the first term ad commo ratio. a 3, 15, 75, 375, 1875, b 7, 13, 25, 49, 97, c 8, 24, 72, 216, 648, d 128, 32, 8, 2, 1 2,... e 2, 6, 12, 2, 3, f 3, 3 3, 9, 9 3, 27, What is the value of x i the followig geometric sequeces? a x, 14, 28, c x + 1, 3x + 3, 1x + 5, b 2x, 4x, 8 + 6x, 15 a Fid the first four terms of the geometric sequece where the 6th term is 243 ad the 8th term is b Fid the first four terms of the geometric sequece where the 3rd term is 331 ad the 5th term is Fid the values of the 2d ad 3rd terms of the geometric sequece show i the followig graph. 8 t 7 (1, 72) (4, 9) (5, 4.5) Term value Term umber 17 A geometric sequece has a 1st term of 2 ad a 6th term of Idetify the values of the 2d, 3rd, 4th ad 5th terms. 18 The umber of ats i a coloy doubles every week. If there are 2944 ats i the coloy at the ed of 8 weeks, how may ats were i the coloy at the ed of the first week? 19 Joas starts a ew job with a salary of $55 per year ad the promise of a 3% pay rise for each subsequet year i the job. a Write a equatio to determie Joas salary i his th year i the job. b How much will Joas ear i his 5th year i the job? 2 The 1st term of a geometric sequece is 13 ad the 3rd term of the same sequece is 117. a Explai why there are two possible values for the commo ratio of the sequece. b Calculate both possible values of the 6th term of the sequece. 21 Julio s parets ivest $5 ito a college fud o his 5th birthday. The fud pays a compoud iterest rate of 5.5% p.a. How much will the fud be worth whe Julio turs 18? 22 A meteoroid is burig up as it passes through the Earth s atmosphere. For every 5 km it travels, the mass of the meteoroid decreases by 5%. At the start of its Topic 6 Sequeces 235

27 descet ito the Earth s atmosphere, at 1 km above groud level, the mass of the meteoroid is 675 g. a Formulate a equatio to determie the mass of the meteoroid after each 5-km icremet of its descet. b What is the mass of the meteoroid whe it hits the Earth, correct to 2 decimal places? Master 6.4 Uits 1 & 2 AOS 3 Topic 3 Cocept 2 Liear recurrece relatios Cocept summary Practice questios Iteractivity Iitial values ad first-order recurrece relatios it The umber of pieces of stoe used to build a pyramid decreases i a ratio of 1 3 for each layer of the pyramid. The pyramid has 9 layers. The top (9th) layer of the pyramid eeded oly 2 stoes. a How may stoes were eeded for the base layer of the pyramid? b Write a equatio to express how may stoes were eeded for the th layer of the pyramid. c How may stoes were eeded for the etire pyramid? 24 The populatios of Melboure ad Sydey are projected to grow steadily over the ext 2 years. A govermet agecy predicts that the populatio of Melboure will grow at a steady rate of 2.6% per year ad the populatio of Sydey will grow at a steady rate of 1.7% per year. a If the curret populatio of Melboure is 4.35 millio, formulate a equatio to estimate the populatio of Melboure after years. b If the curret populatio of Sydey is 4.65 millio, formulate a equatio to estimate the populatio of Sydey after years. c Usig CAS, determie how log it will take for the populatio of Melboure to exceed the populatio of Sydey. Give your aswer correct to the earest year. Recurrece relatios Usig first-order liear recurrece relatios to geerate umber sequeces I a recurrece relatio, the terms of a sequece are depedet o the previous terms of the sequece. A first-order liear recurrece relatio is a relatio whereby the terms of the sequece deped oly o the previous term of the sequece, which meas that we eed oly a iitial value to be able to geerate all remaiig terms of the sequece. I a recurrece relatio, the th term is represeted by t, with the term directly after t beig represeted by t + 1 ad the term directly before t beig represeted by t 1. The iitial value of the sequece is represeted by the term t 1. If the iitial value i a recurrece relatio chages, the the whole sequece chages. If we are ot give a iitial value, we caot determie ay terms i the sequece. 236 Maths Quest 11 GENERAL MATHEMATICS VCE Uits 1 ad 2

28 WORKeD example 14 Determie the first five terms of the sequece represeted by the recurrece relatio t = 2t 1 + 5, give that t 1 = 8. THINK WRITE 1 Idetify what the recurrece relatio meas. t = 2t Each term of the sequece is give by multiplyig the previous term of the sequece by 2 ad addig 5 to the result. 2 Use the recurrece relatio to determie the value of t 2. 3 Use the recurrece relatio to determie the value of t 3. 4 Use the recurrece relatio to determie the value of t 4. 5 Use the recurrece relatio to determie the value of t 5. t 2 = 2t = = = 21 t 3 = 2t = = = 47 t 4 = 2t = = = 99 t 5 = 2t = = = 23 6 Write the aswer. The first five terms of the sequece are 8, 21, 47, 99 ad 23. Iteractivity First-order recurrece relatios with a commo differece it-6264 usig a recurrece relatio to geerate arithmetic sequeces If we kow the values of a ad d i a arithmetic sequece, we ca set up a recurrece relatio to geerate the sequece. A recurrece relatio represetig a arithmetic sequece will be of the form t +1 = t + d, t 1 = a. WORKeD example 15 Set up a recurrece relatio to represet the arithmetic sequece 9, 5, 1, 3, 7, THINK WRITE 1 Determie the commo differece by d = 5 9 subtractig the first term from the secod term. = = 4 2 t 1 represets the first term of the sequece. t 1 = 9 3 Set up the recurrece relatio with the give iformatio. t +1 = t + 4, t 1 = 9 Topic 6 SeQueNCeS 237

29 Iteractivity First-order recurrece relatios with a commo ratio it-6263 usig a recurrece relatio to geerate geometric sequeces If we kow the values of a ad r i a geometric sequece, we ca set up a recurrece relatio to geerate the sequece. A recurrece relatio represetig a geometric sequece will be of the form t +1 = rt, t 1 = a. WORKeD example 16 Set up a recurrece relatio to represet the sequece 1, 5, 2.5, 1.25,.625, THINK 1 Determie the commo ratio by dividig the secod term by the first term. WRITE r = t 2 t 1 = 5 1 = t 1 represets the first term of the sequece. t 1 = 1 3 Set up the recurrece relatio with the give iformatio. t +1 = 1 2 t, t 1 = 1 usig recurrece relatios to model practical situatios Whe there is a situatio that ca be modelled by a arithmetic or geometric sequece, we ca use a recurrece relatio to model it. The first step we eed to take is to decide whether the iformatio suggests a arithmetic or geometric sequece. If there is a commo differece betwee terms, we ca use a arithmetic sequece, ad if there is a commo ratio betwee terms, we ca use a geometric sequece. Spottig arithmetic sequeces Arithmetic sequeces are sequeces that ivolve liear growth or decay. Examples iclude simple iterest loas or ivestmets, the reveue from the sale of a certai amout of items of the same price, ad the umber of flowers left i a field if the same amout is harvested each day. Spottig geometric sequeces Geometric sequeces are sequeces that ivolve geometric growth or decay. Examples iclude compoud iterest loas or ivestmets, the reducig height of a boucig ball, ad the umber of bacteria i a culture after x periods of time. If percetages are ivolved i geeratig a sequece of umbers, this ca result i a geometric sequece. Whe there is a percetage icrease of x percet betwee terms, the value of the commo ratio, r, will be 1 + x 1. Similarly, whe there is a percetage decrease of x percet betwee terms, the value of the commo ratio, r, will be 1 x MaThS QueST 11 GeNeRaL MaTheMaTICS VCe uits 1 ad 2

30 WORKeD example 17 Accordig to the Iteratioal Federatio of Teis, a teis ball must meet certai bouce regulatios. The test ivolves the droppig of a ball vertically from a height of 254 cm ad the measurig the reboud height. To meet the regulatios, the ball must reboud 135 to 147 cm high, just over half the origial distace. Jaie decided to test the ball bouce theory out. She dropped a ball from a height of 2 cm. She foud that it bouced back up to 18 cm, with the secod reboud reachig cm ad the third reboud reachig cm. a Set up a recurrece relatio to model the bouce height of the ball. b Use your relatio from part a to estimate the height of the 4th ad 5th rebouds, givig your aswers correct to 2 decimal places. c Sketch the graph of the umber of bouces agaist the height of each bouce. THINK a 1 List the kow iformatio. 2 Check if there is a commo ratio betwee cosecutive terms. If so, this situatio ca be modelled usig a geometric sequece. 3 Set up the equatio to represet the geometric sequece. b 1 Use the formula from part a to fid the height of the 4th reboud ( = 4). 2 Use the formula from part a to fid the height of the 5th reboud ( = 5). WRITE/dRaW a 1st bouce: 18 cm 2d bouce: cm 3rd bouce: cm t 2 t 1 = =.54 t 3 t 2 = = There is a commo ratio betwee cosecutive terms of.54. a = 18 r =.54 t +1 = rt, t 1 = a t +1 =.54t, t 1 = 18 b t 3 = t +1 =.54t t 4 =.54t 3 = = = 17. (correct to 2 decimal places) t +1 =.54t t 5 =.54t 4 = = Write the aswer. The estimated height of the 4th reboud is 17. cm, ad the estimated height of the 5th reboud is 9.18 cm. Topic 6 SeQueNCeS 239

31 c 1 Draw up a table showig the bouce umber agaist the reboud height. c Bouce umber Reboud height (cm) Idetify the poits to be plotted o the graph. The poits to be plotted are (1, 18), (2, 58.32), (3, 31.49), (4, 17.1) ad (5, 9.18). 3 Plot the poits o the graph. Reboud height (cm) 14 t Number of rebouds Uits 1 & 2 AOS 3 Topic 3 Cocept 8 Geeratio ad evaluatio of the Fiboacci sequece Cocept summary Practice questios The Fiboacci sequece I 122, the Italia mathematicia Leoardo Fiboacci itroduced the Wester world to a uique sequece of umbers which we ow call the Fiboacci sequece. The Fiboacci sequece begis with two 1s, ad every subsequet term of the sequece is foud by addig the two previous terms, givig the sequece: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, Geeratig the Fiboacci sequece usig a recurrece relatio Ulike the first-order recurrece relatios that we have previously used to represet sequeces, the Fiboacci sequece depeds o two previous terms, ad is therefore a secod-order recurrece relatio. The Fiboacci sequece ca be represeted by the recurrece relatio F +2 = F + F +1, F 1 = 1, F 2 = 1. Ratio t 2 t 1 = 1 1 The Golde Ratio The ratios betwee cosecutive terms of the Fiboacci sequece is ot a fixed ratio, as with the geometric sequeces we ve studied. However, the ratios do coverge o a umber that has special mathematical sigificace. t 3 t 2 = 2 1 t 4 t 3 = 3 2 t 5 t 4 = 5 3 t 6 t 5 = 8 5 t 7 t 6 = 13 8 t 8 t 7 = t 9 t 8 = t 1 t 9 = Value Maths Quest 11 GENERAL MATHEMATICS VCE Uits 1 ad 2

32 The umber that the ratios coverge to is called the Golde Ratio. It has a exact value of Throughout history may people have believed that the secrets of beauty lie i the Golde Ratio. Leoardo Da Vici drew his picture of the Vitruvia Ma usig the Golde Ratio, ad parts of the face of the Moa Lisa are i the proportios of the Golde Ratio. Graphig the Fiboacci sequece If you plot the graph of the term umbers of the Fiboacci sequece agaist the term values, the patter forms a smooth curve, similar to the graphs of geometric sequeces. 16 F Variatios of the Fiboacci sequece The stadard Fiboacci sequece begis with two 1s, which are used to formulate the rest of the sequece. If we chage these two startig umbers, we get alterative versios of the Fiboacci sequece. For example, if the first two umbers are 2 ad 5, the sequece becomes: 2, 5, 7, 12, 19, 31, 5, 81, 131, WORKeD example 18 State the first 8 terms of the variatio of the Fiboacci sequece give by the recurrece relatio F +2 = F + F +1, F 1 = 1, F 2 = 5. THINK WRITE 1 State the kow terms. F 1 = 1, F 2 = 5 2 Use the recurrece relatio to geerate the remaiig terms. F +2 = F + F +1 F 3 = F 1 + F 2 = = 4 F 4 = F 2 + F 3 = = 9 F 5 = F 3 + F 4 = = 13 F 6 = F 4 + F 5 = = 22 Topic 6 SeQueNCeS 241

33 F 7 = F 5 + F 6 = = 35 F 8 = F 6 + F 7 = = 57 3 Write the aswer. The first 8 terms of the sequece are 1, 5, 4, 9, 13, 22, 35, 57. Exercise 6.4 PRactise Recurrece relatios 1 WE14 Determie the first five terms of the sequece represeted by the recurrece relatio t =.5t 1 + 8, give that t 1 = Determie the first five terms of the sequece represeted by the recurrece relatio t = 3t 1 4, give that t 1 = 2. 3 WE15 Set up a recurrece relatio to represet the arithmetic sequece 2, 3, 8, 12, A arithmetic sequece is represeted by the recurrece relatio t +1 = t + 3.5, t 1 = 2.2. Determie the first 5 terms of the sequece. 5 WE16 Set up a recurrece relatio to represet the geometric sequece 2.5, 7.5, 22.5, 67.5, 22.5, 6 A geometric sequece is represeted by the recurrece relatio t +1 = 3.5t, t 1 = 4. Determie the first five terms of the sequece. 7 WE17 Eric decided to test the reboud height of a teis ball. He dropped a ball from a height of 3 cm ad foud that it bouced back up to 165 cm, with the secod reboud reachig 9.75 cm, ad the third reboud reachig cm. a Set up a recurrece relatio to model the bouce height of the ball. b Use your relatio from part a to estimate the height of the 4th ad 5th rebouds, givig your aswers correct to 2 decimal places. c Sketch the graph of the umber of bouces agaist the height of the bouce 8 Rosaa decided to test the ball reboud height of a basketball. She dropped the basketball from a height of 5 cm ad oted that each successive reboud was two-fifths of the previous height. a Set up a recurrece relatio to model the bouce height of the ball. b Use your relatio to estimate the heights of the first 5 rebouds, correct to 2 decimal places. c Sketch the graph of the first 5 bouces agaist the reboud height. 9 WE18 State the first 8 terms of the variatio of the Fiboacci sequece give by the recurrece relatio F +2 = F + F +1, F 1 = 3, F 2 = 5. 1 The Lucas sequece is a special variatio of the Fiboacci sequece that starts with the umbers 2 ad 1. Determie the first 1 umbers of the Lucas sequece. 242 Maths Quest 11 GENERAL MATHEMATICS VCE Uits 1 ad 2

34 Cosolidate 11 The 3rd ad 4th terms of a arithmetic sequece are 7 ad Set up a recurrece relatio to defie the sequece. 12 The 4th ad 5th terms of a geometric sequece are 4 ad 1. Set up a recurrece relatio to defie the sequece. 13 A variatio of the Fiboacci sequece has a 3rd term of 2 ad a 4th term of 6. Determie the recurrece relatio for this sequece. 14 Brett ivests $18 i a accout payig simple iterest. After 3 moths he has $ i his accout. a Set up a recurrece relatio to determie the amout i Brett s accout after moths. b How much will Brett have i his accout after 7 moths? 15 Graph the first 7 terms of the variatio of the Fiboacci sequece that starts with the umbers 3 ad Cassadra has $6615 i her bak accout after 2 years ad $ i her bak accout after 3 years. Her accout pays compoud iterest. a Set up a recurrece relatio to determie the amout i Cassadra s accout after years. b How much does Cassadra have i her accout after 5 years? 17 A ice shelf is shrikig at a rate of 12 km 2 per year. Whe measuremets of the ice shelf bega, the area of the shelf was 37 km 2. a Create a recurrece relatio to express the area of the ice shelf after years. b Use your relatio to determie the area of the ice shelf after each of the first 6 years. c Plot a graph showig the area of the shrikig ice shelf over time. 18 The umber of bacteria i a coloy is icreasig i lie with a secod-order recurrece relatio of the form t +2 = 2t + t +1, t 1 = 3, t 2 = 9, where is the time i miutes. Determie the amout of bacteria i the coloy after each of the first 1 miutes. Topic 6 Sequeces 243

35 19 A boucig ball rebouds to 7% of its previous height. a From how high would the ball have to be dropped for the 1th bouce to reach 5 cm i height? Give your aswer correct to 1 decimal place. b Defie a recurrece relatio to determie the height of the ball after bouces. 2 A abadoed islad is slowly beig overru with rabbits. The populatio of the rabbits is approximately followig a Fiboacci sequece. The estimated umber of rabbits after 4 years of moitorig is 35, ad the estimated umber after 5 years of moitorig is 55. a Estimate the umber of rabbits after the first year of moitorig. b Create a recurrece relatio to determie the umber of rabbits after years. c Is it realistic to expect the populatio of rabbits to cotiue to icrease at this rate? Explai your aswer. Master 21 Luke ad Lucida are sibligs who are give $3 to ivest by their parets. Luke ivests his $3 i a simple iterest bod payig 4.8% p.a., ad Lucida ivests her $3 i a compoud iterest bod payig 4.3% p.a. a Write a recurrece relatio to express the amout i Luke s accout after years. b Write a recurrece relatio to express the amout i Lucida s accout after years. c Determie the amout i each of their accouts for the first 7 years. d Draw a graph showig the amout i each accout over the first 7 years. 22 Variatios of the Fiboacci sequece will always ted towards plus or mius ifiity. By alterig the two startig umbers of the Fiboacci sequece, determie whether this statemet is true or ot. 244 Maths Quest 11 GENERAL MATHEMATICS VCE Uits 1 ad 2

36 ONLINE ONLY 6.5 Review The Maths Quest Review is available i a customisable format for you to demostrate your kowledge of this topic. The Review cotais: Multiple-choice questios providig you with the opportuity to practise aswerig questios usig CAS techology short-aswer questios providig you with the opportuity to demostrate the skills you have developed to efficietly aswer questios usig the most appropriate methods Exteded-respose questios providig you with the opportuity to practise exam-style questios. a summary of the key poits covered i this topic is also available as a digital documet. REVIEW QUESTIONS Dowload the Review questios documet from the liks foud i the Resources sectio of your ebookplus. ONLINE ONLY Activities To access ebookplus activities, log o to Iteractivities A comprehesive set of relevat iteractivities to brig difficult mathematical cocepts to life ca be foud i the Resources sectio of your ebookplus. studyon is a iteractive ad highly visual olie tool that helps you to clearly idetify stregths ad weakesses prior to your exams. You ca the cofidetly target areas of greatest eed, eablig you to achieve your best results. Uits 1 & 2 Sequeces Sit topic test Topic 6 SeQueNCeS 245