6.1. Sequences as Discrete Functions. Investigate

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1 6.1 Sequeces as Discrete Fuctios The word sequece is used i everyday laguage. I a sequece, the order i which evets occur is importat. For example, builders must complete work i the proper sequece to costruct safe, strog houses. I mathematics, a sequece is a set of umbers, usually separated by commas, arraged i a particular order. Some sequeces have very specific patters ad ca be represeted by mathematical rules or fuctios. May atural pheomea, such as the spiral patters see i seashells, suflowers, ad galaxies, ca be represeted by sequeces. sequece a ordered list of umbers idetified by a patter or rule that may stop at some umber or cotiue idefiitely 3, 7, 11, 15, 6, 18, 5, a fuctio whose domai is the set, or a subset, of the atural umbers ad whose rage is the terms of the sequece Tools square dot paper ruler Ivestigate How ca you relate each umber of a sequece to its positio i the list? The patter of floor tiles i a ew hotel is desiged usig ested squares. Explore the sequece formed by the umber of eclosed regios created. Method 1: Use Pecil ad Paper 1. Costruct a 16-uit by 16-uit square.. Locate the midpoit of each side. Coect cosecutive midpoits to form a ew square. Cotiue costructig midpoits ad smaller squares util the squares are too small to work with. 3. Copy ad complete the table for the patter you created. Number of Squares, Number of Regios, t MHR Fuctios 11 Chapter 6

2 . Refer to your completed table. a) Reflect Describe the patter i the Number of Regios colum. b) Write the values i this colum as a sequece of terms. Use the patter to write the ext three terms of the sequece. 5. Graph the sequece usig ordered pairs of the form (umber of squares, umber of regios). Should you joi the poits with a smooth curve or lie or leave them as distict poits? Explai your thikig. 6. Reflect Determie a explicit formula to describe the umber of regios,, accordig to the umber of squares,. term (of a sequece) a sigle value or object i a sequece explicit formula a formula that represets ay term i a sequece relative to the term umber,, where Method : Use The Geometer s Sketchpad 1. Ope The Geometer s Sketchpad. From the Graph meu, choose Grid Form ad the Rectagular Grid. Right-click o each axis ad choose Hide Axis from the drop-dow meu. Tools computer with The Geometer s Sketchpad. From the Graph meu, choose Sap Poits. Draw a 16-uit by 16-uit square. 3. Select the sides of the square. From the Costruct meu, choose Midpoits. Joi the midpoits with lie segmets to form a square. Cotiue costructig midpoits ad smaller squares util the squares are too small to work with.. Copy ad complete the table for the patter you created. Number of Squares, Number of Regios, t Refer to your completed table. a) Reflect Describe the patter i the Number of Regios colum. b) Write the values i this colum as a sequece of terms. Use the patter to write the ext three terms of the sequece. 6. Graph the sequece usig ordered pairs of the form (umber of squares, umber of regios). Should you joi the poits with a smooth curve or lie or leave them as distict poits? Explai your thikig. 7. Reflect Determie a explicit formula to describe the umber of regios,, accordig to the umber of squares,. 6.1 Sequeces as Discrete Fuctios MHR 355

3 Example 1 Use the Explicit Formula to Write Terms i a Sequece Write the first three terms of each sequece, give the explicit formula for the th term of the sequece,. a) b) 5 1 Techology Tip See the Use Techology feature at the ed of this sectio for a TI-Nspire graphig calculator solutio. Techology Tip Whe usig the sequece fuctio of a graphig calculator to geerate the terms of a sequece, you eed to specify five thigs. For example, to geerate the first three terms of the sequece = 3 1, you eed the expressio for the xth term of the sequece, 3x 1 the variable, x the startig term umber, 1 the edig term umber, 3 the icremet value for the term umbers, 1 Solutio a) Method 1: Use Pecil ad Paper To determie the value of the first three terms, substitute the term umbers 1,, ad 3 for t 1 5 3(1) 1 t 5 3() 1 t 3 5 3(3) The first three terms of the sequece are, 11, ad 6. Method : Use a Graphig Calculator The terms of a sequece ca be geerated usig a graphig calculator. Press d [LIST] ad cursor over to the OPS meu. Select 5:seq( ad eter 3x 1, x, 1, 3, 1). Press ENTER. The first three terms of the sequece are, 11, ad 6. b) Substitute the term umbers 1,, ad 3 for. 5 1 t t _ The first three terms of the sequece are 0, 1_ Example t _ 3, ad _ 3. Determie Explicit Formulas i Fuctio Notatio For each sequece, make a table of values usig the term umber ad term ad calculate the fiite differeces. The, graph the sequece usig the ordered pairs (term umber, term) ad determie a explicit formula for the th term, usig fuctio otatio. a) 7, 1, 17,, b) 1, 10, 5, 6, 356 MHR Fuctios 11 Chapter 6

4 Solutio a) Patters i fiite differeces tables ca be used to help determie a formula for the terms i a sequece. Term Number, Term, First Differeces The graph models a fuctio sice there is exactly oe value of for each value of. This fuctio is liear sice the first differeces are costat. The rate of chage, or the slope, is the first differece, 5. f () 5 5 b By ispectio, b 5. The, a explicit formula to determie the terms i the sequece is f () 5 5, where N. b) Fid the first ad secod differeces. Term Number, Term, First Differeces Secod Differeces Sice the secod differeces are costat, this fuctio is quadratic. Half the value of the secod differece correspods to the value of a i a quadratic fuctio of the form f () 5 a b c. f () 5 3 b c To determie the values of b ad c, substitute the coordiates of two poits ad solve a liear system of equatios. For (1, 1), 1 5 3(1) b c, or 5 b c. For (, 10), () b c, or 5 b c. 5 b c 5 b c b 1 b 5 0 Substitute b 5 0 ito equatio 1 ad solve for c. 5 b c 5 0 c c 5 0 A explicit formula to determie the terms i the sequece is f () 5 3, where. 6.1 Sequeces as Discrete Fuctios MHR 357

5 Example 3 Types of Fuctios cotiuous fuctio a fuctio that maps real umbers to real umbers ad has a graph that is a curve with o holes or jumps discrete fuctio a fuctio whose graph is made up of separate poits that are ot coected a) The charge i a battery decreases by about % per day ad ca be modelled by the fuctio C(d) 5 100(0.98) d, where d is the time, i days, ad C is the level of the charge, as a percet. How much charge is left after 10 days? Is this a cotiuous fuctio or a discrete fuctio? Explai. b) A certai bacterial culture starts with 00 bacteria ad doubles every hour. Its growth ca be modelled by the fuctio N(t) 5 00() t, where t is the time, i hours, ad N is the umber of bacteria. How may bacteria will there be after 10 h? Is this fuctio cotiuous or discrete? Explai. Solutio a) To determie the charge level after 10 days, substitute d 5 10 ito C(d) 5 100(0.98) d. C(10) 5 100(0.98) After 10 days, the charge level of the battery is approximately 81.7%. The table of values ad graph show how the battery charge chages over time. Time (days), d Charge Level (%), C Charge Level (%) C This is a cotiuous fuctio sice the charge level will be cotiuously chagig over time. It does ot drop % suddely at the ed of every day, but gradually decreases as time goes by. 0 d Time (days) b) To determie the umber of bacteria after 10 h, substitute t 5 10 ito N(t) 5 00() t. N(10) 5 00() After 10 h, there will be bacteria. The table of values ad graph show the umber of bacteria over time. 358 MHR Fuctios 11 Chapter 6

Time (h), t Number of Bacteria, N 0 00 1 00 800 3 1600 300 Number of Bacteria N 000 000 0 Time (h) t Sice you caot have part of a bacterium, this fuctio is discrete.

6 Time (h), t Number of Bacteria, N Number of Bacteria N Time (h) t Sice you caot have part of a bacterium, this fuctio is discrete. After each hour, the umber of bacteria is double the umber of the previous hour. Example Determie the Value of a Car The value of a ew car purchased for $ depreciates at a rate of 15% i the first year ad % every year after that. a) Determie the value of the car at the ed of the first year, the secod year, ad the third year. Write these values as a sequece. Coectios Whe a item depreciates, its value decreases over time. Some examples of items that depreciate are vehicles, electroics, computers, ad clothig. b) Determie a explicit formula for the value of the car at the ed of year. c) What is the value of the car at the ed of year 0? Is this realistic? Explai your thikig. Solutio a) The value of the car whe it is ew is $ At the ed of the first year, the car depreciates by 15%, so it is worth $ , or $ At the ed of the secod year, the car value decreases by %, so it is worth 96% of its value at the start of that year $ $63 68 At the ed of year 3, the car value agai decreases by % (0.96 $66 300) $ $ The sequece that represets the value of the car at the ed of each year is , 63 68, , or , 0.96(66 300), 0.96 (66 300),. b) The explicit formula for the value of the car at the ed of year is (0.96) Sequeces as Discrete Fuctios MHR 359

c) Substitute 5 0 to fid the value of the car at the ed of year 0. t 5 66 300(0.96) 1 t0 5 66 300(0.96)0 1 5 66 300(0.96)19 30 55.79

7 c) Substitute 5 0 to fid the value of the car at the ed of year 0. t (0.96) 1 t (0.96) (0.96) At the ed of year 0, the value of the car is $ It is possible that the car will be worth more tha this if the ower looks after it carefully. It could also be worth much less if it has bee ivolved i a collisio, has a high odometer readig, or has a lot of rust. There are may factors that ca affect the value of a used car. Key Cocepts A sequece of umbers ca be represeted by a discrete fuctio. The graph of a discrete fuctio is a distict set of poits, ot a smooth curve. The domai of a fuctio represetig a sequece is the set or a subset of the atural umbers, N. Give the explicit formula for the th term, t or f (), of a sequece, the terms ca be writte by substitutig the term umbers for. Examples of explicit formulas are t 5 3 ad f () A explicit formula for the th term of a sequece ca sometimes be determied by fidig a patter amog the terms. Commuicate Your Uderstadig C1 Graph the sequece of umbers represeted by the ordered pairs (1, 1), (, 1), (3, 3), (, 5),. O the same set of axes, graph the fuctio f (x) 5 x 3, x R. Describe the similarities ad differeces betwee the two graphs. Write the formula for the sequece usig fuctio otatio ad specify the domai. C Cosider the domai of a cotiuous ad of a discrete fuctio. What are the similarities betwee the domais? How are they differet? C3 Describe two situatios i which it might be importat to kow a specific term i a sequece. A Practise For help with questios 1 ad, refer to Example Write the first three terms of each sequece, give the explicit formula for the th term of the sequece. a) t b) t 5 5 c) t d) f () 5 e) t f) f () 5 3() Write the 1th term, give the explicit formula for the th term of the sequece. a) f () b) t 5 5 c) f () 5 d) t 5 e) t 5 f) f () 5 ( ) MHR Fuctios 11 Chapter 6 Fuctios 11 CH06.idd 360 6/10/09 :19:36 PM

8 For help with questios 3 to 5, refer to Example. 3. Describe the patter i each sequece. Write the ext three terms of each sequece. a), 16, 6, 56, b) 7, 6, 5,, c) 3, 6, 9, 1, d) 100, 10, 1, 0.1, e) 5, 10, 15, 0, f) 1_, 1_, 1_, 1_, g) x, 3x, 5x, 7x, h), 8, 1, 16, i) a, ar, ar, ar 3, j) 0., 0., 0.6, 0.8, c) d) 0 1 For help with questio 6, refer to Example For each graph, specify whether the fuctio is discrete or cotiuous ad explai your choice. a) t For each sequece, make a table of values usig the term umber ad term ad calculate the fiite differeces. The, determie a explicit formula i fuctio otatio ad specify the domai. a),, 6, 8, b), 1, 0, 1, c) 3, 6, 9, 1, b) 0 y 0 x d) 0, 3, 8, 15, e) 3, 6, 11, 18, c) y f) 10, 9, 0, 17, 5. The graphs show the terms i a sequece. Write each sequece i fuctio otatio ad specify the domai. a) b) 0 x B Coect ad Apply 7. Describe the patter i each sequece ad write the ext three terms. 0 0 a) 1, 1, 1,, 1, 3, 1,, 1, b) 1, 5,, 10, 3, 15, c) 3, 3 5, 15, 15 5, 75,... d) 1_, 1_, 1_, 1_, Sequeces as Discrete Fuctios MHR 361

9 8. Cosider the sequece 7, 1, 1, 8,. Determie whether or ot each of the followig umbers is part of this sequece. Explai your thikig. a) 98 b) 110 c) 378 d) Use Techology The world Represetig populatio i 1995 was 5.7 billio. Coectig Sice the the growth rate has bee approximately 1.% per year. a) Graph the equatio y 5 5.7(1.01) x usig a graphig calculator with the widow settigs show. b) Describe the shape of the graph. How would the graph chage if the growth rate were greater? Use the graphig calculator to verify your descriptio. c) Assume the tred cotiues. Determie the populatio i each year from 007 to 015. Write these umbers as a sequece. 10. Use Techology A ew car valued at $ will depreciate at a average rate of 0% per year over the ext several years. a) Eter the followig iformatio i a spreadsheet. A B 1 Year Value =A+1 =0.8*B Reasoig ad Provig Problem Solvig Commuicatig b) Use Fill Dow to calculate the value of the car for the ext 15 years. c) Make a XY (Scatter) plot of this data. d) Use fuctio otatio to write a explicit formula to represet the value of the car at the ed of year. e) Is this a cotiuous or a discrete fuctio? Explai your thikig. Selectig Tools Reflectig 11. If two resistors, A ad B, are i parallel, the the combied resistace, R, i ohms (Ω), is foud by the formula 1_ R 5 1_ 1_ R A R B Assume each resistor has a resistace of 1 Ω. Use the formula to determie the value of the resistace for, 3,, 5, ad 6 resistors i parallel. Write these umbers as a sequece. 1. Chapter Problem The Koch sowflake was oe of the earliest fractals to be described. The sowflake starts as a equilateral triagle. At each stage, the middle third of each side is replaced by two lie segmets, each equal i legth to the lie segmet they replace. Stage 1 Stage Stage 3 a) Work with a parter ad use isometric dot paper to draw the diagrams show. Use the patter to draw the ext diagram. b) Copy ad complete the table. Stage Number Lie Segmet Legth Number of Lie Segmets Perimeter of the Sowflake _ 3 1 1_ c) Determie a explicit formula for the th term i colums two, three, ad four of the table. d) Use your formulas to calculate the values for stage. A B 36 MHR Fuctios 11 Chapter 6

10 13. Determie a explicit formula for the th term of each sequece. Use the formula to write the 15th term. a), 8, 16, 3, b) 1, _, 3_, _ 3 5 7, c) 1,, 3,,... d) 1,,, 8, e) 1, 1_, 1_ 3, 1_,... f) 1, 1, 1, 1, 1. Create two differet sequeces that start with 1,, 3. Write a explicit formula for the th term of each sequece i fuctio otatio. Graph each sequece. 15. A ew small busiess plas to double its sales every day for its first weeks. Sales o the first day are $50. a) Write the sequece that represets the sales for the first 6 days accordig to the pla. b) Write a explicit formula to determie the sales o ay of the first 1 days. c) Use your formula to determie the sales o the 1th day. Is this reasoable? Why or why ot? 16. A high school is experiecig decliig erolmet. This year the erolmet was 100, ad it has bee predicted that every year there will be 110 fewer studets. Write a explicit formula to determie the umber of studets i ay give year. After how log will the erolmet drop below 800 studets? C Exted Represetig Coectig Reasoig ad Provig Problem Solvig Commuicatig Selectig Tools Reflectig 17. a) Use a calculator to determie approximate values for the first three terms of the sequece 3, 3, 3, 3,... b) Describe the patter i the sequece. c) Use the patter to predict the value of the 50th term i this sequece. 18. Determie a explicit formula for the total umber of squares i a by square. 1 square 5 squares 1 squares 19. Math Cotest The multiples of 5 are prited i the colums of a Bigo game card as show. B I N G O If the umbers cotiue i this patter, i which colum will the umber 5555 occur? A B B N C G D O 0. Math Cotest The crow jewels are missig. Scotlad Yard has four suspects, Albert, Bob, Cecilia, ad Dwight. Albert says, Cecilia is the thief. Bob says, I am ot the thief. Cecilia says, Dwight is the thief. Dwight says, Cecilia lied. If oly oe of these statemets is true, who is the real thief? A Albert B Bob C Cecilia D Dwight 1. Math Cotest A bag cotais two balls. The value of oe ball is. The value of the other is 9. A ball is chose ad the value of the ball is added to a ruig total. The umber of differet sums that are ot possible to attai is A 6 B 1 C 11 D Not possible to determie 6.1 Sequeces as Discrete Fuctios MHR 363

11 Use Techology Tools TI-Nspire CAS graphig calculator Coectios Example 1 o page 356 is used to model the steps eeded to fid the first three terms of a sequece, give the explicit formula for the th term of the sequece. Techology Tip To type a uderscore (_), use the symbol palette. Press / k. Use the cursor keys to move to _. Press. Use a TI-Nspire CAS Graphig Calculator to Write Terms i a Sequece Write the first three terms of each sequece, give the explicit formula for the th term of the sequece,. A: B: 5 1 Solutio Ope a ew documet. Ope a page usig the Lists & Spreadsheet applicatio. Eter the headigs for colums A, B, ad C. At the top of colum A, type ad press. At the top of colum B, type a_t_ ad press. At the top of colum C, type b_t_ ad press. Eter the values 1,, ad 3 for, startig i cell A1. Eter the explicit formulas for the sequeces. I the formula cell for colum B, type 5 3a 1 ad press. I the formula cell for colum C, type 5 (a 1) a ad press. The first three terms for each sequece will be displayed. 36 MHR Fuctios 11 Chapter 6

Essential Question How can you recognize an arithmetic sequence from its graph?

Essential Question How can you recognize an arithmetic sequence from its graph? . Aalyzig Arithmetic Sequeces ad Series COMMON CORE Learig Stadards HSF-IF.A.3 HSF-BF.A. HSF-LE.A. Essetial Questio How ca you recogize a arithmetic sequece from its graph? I a arithmetic sequece, the