Recursive Procedures. How can you model the relationship between consecutive terms of a sequence?

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1 6. Recursive Procedures I Sectio 6.1, you used fuctio otatio to write a explicit formula to determie the value of ay term i a Sometimes it is easier to calculate oe term i a sequece usig the previous terms. Computer programmers use sequeces of code to create istructios for computers. Ofte these sequeces tell the computer to use a previous value to fid the ext oe. This is kow as a recursive procedure. Ivestigate How ca you model the relatioship betwee cosecutive terms of a sequece? The first three diagrams i a patter are show. Model the patter. Tools grid paper Optioal computer with The Geometer s Sketchpad 1. a) For Diagram 1, draw a square with side legth 1 uit. Diagram 1 b) For Diagram, start with Diagram 1 ad draw a square with side legth 1 uit adjacet to the first square. This creates a rectagle. c) For Diagram 3, draw a larger rectagle. Start with Diagram ad draw a square with side legth uits adjacet to ad directly above the two smaller squares.. Diagram will cotai a square with side legth 3 uits. Where should this square be draw i order to cotiue the patter? Draw Diagram. 3. Copy ad complete the table for the patter of diagrams. Diagram Diagram 3 Diagram Number Side Legth of Square (uits) Recursive Procedures MHR 365

2 . a) Write the side legths of the squares as a b) Reflect Determie the relatioship betwee cosecutive terms i the sequece ad write a formula for the th term, t, i terms of the ( 1)th term, t 1, ad the ( )th term, t. Fiboacci sequece the sequece of umbers 1, 1,, 3, 5, 8, Each umber, after the first two umbers, is the sum of the precedig two umbers. recursio formula a formula by which each term of a sequece is geerated from the precedig term or terms Coectios The Fiboacci sequece is amed after Leoardo Fiboacci (c ), who discovered the sequece while studyig the reproductive ature of rabbits. Fiboacci is also credited with itroducig the decimal umber system to Europe. Go to the Fuctios 11 page o the McGraw-Hill Ryerso Web site ad follow the liks to Chapter 6 to lear more about the Fiboacci The sequece formed by the side legths of the squares i the Ivestigate is a famous sequece kow as the Fiboacci A sequece is said to be recursive if a ew term is foud usig a previous term or terms. For example, the Fiboacci sequece 1, 1,, 3, 5, 8, 13, is a recursive sequece because each term begiig with the third term is the result of addig the two previous terms. The recursio formula for this sequece ca be writte as t 1 5 1, t 5 1, t t, where t 1 is the first term, t is the secod term, t is the th term, ad so o. Example 1 Write the Terms of a Sequece Give the Recursio Formula Write the first four terms of each a) t 1 b) f (1) 5 1_, f () 5 f ( 1) 3_ Solutio a) The first term is give as 3. Use the equatio t to determie the ext three terms i the t 5 t 1 t 3 5 t t 5 t The first four terms of the sequece are 3, 1, 1, 3. b) The first term is give as 1_. Use the equatio f () 5 f ( 1) 3_ to determie the ext three terms i the f () 5 f (1) 3_ f (3) 5 f () 3_ f () 5 f (3) 3_ 5 1_ 3_ 5 1 3_ 5 5_ 3_ _ 5 The first four terms of the sequece are 1_, 1, 5_,. 366 MHR Fuctios 11 Chapter 6

3 Example Write a Recursio Formula Determie a recursio formula for each a) 3, 6, 1,, b) f() c) 3, 5, 8, 1, Solutio a) Look for a patter i the terms. t t 5 t 1 ( ) t 3 5 t ( ) t 5 t 3 ( ) The recursio formula is t 1 5 3, t 5 t 1. b) Look for a patter i the y-coordiates. f (1) 5 f () 5 6 f (3) 5 10 f () 5 1 Each term is more tha the previous term. The recursio formula is f (1) 5, f () 5 f ( 1). c) Look for a patter i the terms. t t 5 t 1 t 3 5 t 3 t 5 t 3 The recursio formula is t Recursive Procedures MHR 367

4 Example 3 Medicatio i the Huma Body A ruer ijures her kee i a race. Her doctor prescribes physiotherapy alog with 500 mg of a ati-iflammatory medicie every h for 3 days. The half-life of the ati-iflammatory medicie is approximately h. This meas that after h, about half of the medicie is still i the body. a) Make a table of values showig the amout of medicie remaiig i the body after each -h period of time. b) Write the amout of medicie remaiig after each -h period as a Write a recursio formula for the c) Graph the d) Describe what happes to the medicie i the ruer s body over time. Solutio Techology Tip To calculate the amout of medicie i the body usig a spreadsheet, isert the formula =0.5*B+500, ad the fill dow. a) This ca be doe usig pecil ad paper or a spreadsheet. Roud your aswers to the earest teth of a milligram, where ecessary. -h Iterval Amout of Medicie (mg) _ 1 (500) = 750 1_ (750) = b) The sequece represetig the amout of medicie remaiig i the body after each -h period is 500, 750, 875, 937.5, 968.8,, The recursio formula is t , t t MHR Fuctios 11 Chapter 6

5 Amout of Medicie (mg) c) Amout of Medicie i the Body h Iterval 1 d) From the graph, the amout of medicie i the body icreases util it appears to reach a costat level of about 1000 mg. Key Cocepts A recursive procedure is oe where a process is performed o a iitial object or umber ad the the result is put through the steps of the process agai. This is repeated may times over. A sequece ca be defied recursively if each term ca be calculated from the previous term or terms. A recursio formula shows the relatioship betwee the terms of a A sequece ca be represeted by a patter, a explicit formula, or a recursio formula. Formulas ca also be writte usig fuctio otatio. For example: Patter: 1, 3, 5, 7, Explicit formula: t 5 1 or f () 5 1 Recursio formula: t1 5 1, t 5 t 1, or f (1) 5 1, f () 5 f ( 1) I a explicit or a recursio formula for a sequece, is a atural umber because it is a term umber. To fid the terms of a sequece usig a recursio formula, begi with the ext atural umber that is ot used i the formula. Commuicate Your Uderstadig C1 What do you eed to kow about a sequece i order to write a recursio formula to describe the terms i the sequece? C a) The recursio formula t1 5 5, t 5 t 1 1, has two parts. Describe the two parts. b) Why does a recursio formula have at least two parts? c) What characteristic of a sequece is eeded for the formula to have more tha two parts? C3 A sequece has the recursio formula t1 5, t 5 t 1 5. Use words to describe how this formula is used to determie cosecutive terms i the C The explicit formula for the th term of a sequece is t 5 6( 1) 1, while the recursio formula for the same sequece is t1 5 5, t 5 t 1 1. Whe might it be more coveiet to use oe form of the formula istead of the other? Explai. 6. Recursive Procedures MHR 369 Fuctios 11 CH06.idd 369 6/10/09 :19:57 PM

6 A Practise For help with questios 1 ad, refer to Example Write the first four terms of each sequece, where N. a) t 1 5, t 3 b) t 1 5 7, t 1 c) t 1 5 3, t 5 0.t 1 1. _ d) t , t 5 t 1 e) t 1 5 8, t 5 3t 1 _ f) t , t 5 5t Write the first four terms of each sequece, where N. a) f (1) 5 9, f () 5 f ( 1) b) f (1) 5 1, f () 5 3f ( 1) f ( 1) c) f (1) 5 3, f () 5 d) f (1) 5 18, f () 5 f ( 1) e) f (1) 5 0.5, f () 5 f ( 1) f) f (1) 5 5, f () 5 0.5f ( 1) For help with questios 3 ad, refer to Example. 3. Determie a recursio formula for each a) 5, 11, 17, 3, b), 1,, 5, c), 8, 16, 3, d),, 1, 1_, e) 5, 15, 5, 135,. For each graph, write the sequece of terms ad determie a recursio formula usig fuctio otatio. a) f() b) f() A example of a costat sequece is 06, 06, 06,. Write a recursio formula for this Write aother costat sequece ad its recursio formula. B Coect ad Apply For help with questios 6 ad 7, refer to Example A ew theatre is beig built for a youth orchestra. This theatre has 50 seats i the first row, 5 i the secod row, 6 i the third row, 7 i the ext row, ad so o. a) Represet the umber of seats i the rows as a b) Describe the patter i the umber of seats per row. c) Write a recursio formula to represet the umber of seats i ay row. 7. Sacha ad Marghala paid $ for their first home. The real estate aget told them that the house will appreciate i value by 3% per year. a) Copy ad complete the table to show the value of the house for the ext 10 years. Year House Value ($) = b) Write the value of the house for the first 10 years as a c) Write a recursio formula to represet the value of the house. Use your formula to predict the value after 15 years. 8. Write the first four terms of each a) t 1 5 1, t 5 (t 1 ) 3 f ( 1) b) f (1) 5 8, f () 5 c) t 1 d) t 1 5 5, t 5 t 1 e) t 1 5 1_, t 5 t 1 f) f (1) 5 a 3b, f () 5 f ( 1) b 370 MHR Fuctios 11 Chapter 6

7 9. Use the give recursio formula to determie the first four terms of each The, use words to describe the rule for determiig terms i the a) f (1) 5, f () 5, f () 5 f ( 1) f ( ) b) f (1) 5 1, f () 5, f () 5 f ( 1)f ( ) c) t 1 5 5, t 5 7, t 5 t t 1 d) t 1 5, t 5 3t t 1 e) t 1 5 1, t 5, t 5 t t 1 f) t 1 5 3, t 5 1, t 3 5 7, t 5 t 3 t t The diagrams show the diagoals i regular polygos with sides. Write the sequece for the umber of diagoals ad determie the recursio formula for this 11. A square-based pyramid with height 7 m is costructed with cubic blocks measurig 1 m o each side. Write a recursio formula for the sequece that represets the umber of blocks used at each level from the top dow. 1. A sequece has a first term of 8. Each succeedig term is more tha twice the previous term. a) Write the first four terms of this b) Defie the sequece usig a recursio formula ad the graph the 13. Give the explicit Reasoig ad Provig formula of a Represetig sequece, write the Problem Solvig first four terms ad Coectig the determie a recursio formula Commuicatig for each a) t 5 ( 1) b) t 5 1 c) f () 5 3 d) t e) f () 5 ( )( ) f) f () 5 () 1 1. Give the recursio formula, write the first four terms of the sequece ad the determie the explicit formula for the a) t 1 1 b) t 1 5 1, t 5 1_ t 1 c) t , t 10 d) t 1 5, t 1 ( 1) Reflectig 15. Chapter Problem The Sierpiski triagle was described by Waclaw Sierpiski i This ow famous fractal was also see i Italia art i the 13th cetury. a) Use isometric dot paper or The Geometer s Sketchpad to costruct a large equilateral triagle. Cosider the area of this triagle to be 1 square uit. b) Locate ad joi the midpoits to make a ew triagle. Shade i all but the cetre triagle. Determie the area of the shaded regios. c) Cotiue to draw triagles formed by the midpoits of the sides of the smaller ad smaller shaded triagles. Always leave the cetre triagle ushaded ad determie the area of the shaded regios. d) Write the area of the shaded regios at each stage as a Write a formula for this Is your formula explicit or recursive? Selectig Tools Stage 1 Stage Stage 3 6. Recursive Procedures MHR 371

8 16. Write a recursio formula for each a), 6, 1, 0, 30, b) 3, 7, 16, 3, c), 5, 6, 677, d) 1, 0, 3, 1, Achievemet Check 17. Caadia checkers is played o a checkerboard that has 1 squares per side. What would happe if you placed a pey o the first square, two peies o the secod square, four o the third square, eight o the fourth square, ad so o? a) Write the umbers of peies o the first 1 squares as a b) Write a recursio formula to represet the umber of peies o ay square. Use your formula to determie the umber of peies o square 0. c) Write a explicit formula, i fuctio otatio, to represet the umber of peies o ay square. Use this formula to verify your aswer to part b). d) Is this fuctio discrete or cotiuous? Explai. C Exted 18. Write the first five terms of each sequece, startig at f (1). a) f () 5 3, f () 5 f ( 1) b) f (3) 5 9, f () 5 f ( 1) 19. Create three differet Reasoig ad Provig sequeces that start with, 3,. Write recursio formulas Represetig Problem Solvig for these sequeces. Coectig Reflectig Write the ext two Commuicatig terms of each of your sequeces ad challege your classmates to determie the recursive rule that you used. Selectig Tools 0. a) Write a ew sequece usig the terms of the Fiboacci Start with the first two terms of the Fiboacci The, divide the third term i the Fiboacci sequece by the secod term to get the ext term i the ew Cotiue dividig the ext term i the Fiboacci sequece by the previous term to get the ext term i the ew Describe the patter i the ew b) The sequece i part a) coverges to a value close to This is the umber φ (phi), which is also kow as the golde ratio. Research the golde ratio ad prepare a poster for the classroom. 1. The golde spiral is costructed from the rectagle diagram you costructed i the Ivestigate. a) I each square, costruct a arc with radius equal to the side legth of the square. b) Determie the area uder the spiral for each square. Write the areas as a sequece of umbers. Calculate the total area uder the spiral.. Math Cotest The Lucas umbers are similar to the Fiboacci umbers but the first two terms are t 1 5 ad t 5 1. The sequece is, 1, 3,, 7, 11,. Let L be the th Lucas umber ad F be the th Fiboacci umber. For, show that L F 5 F. 37 MHR Fuctios 11 Chapter 6

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