Chapter 7. In the questions below, describe each sequence recursively. Include initial conditions and assume that the sequences begin with a 1.

Transcription

1 Use the followig to aswer questios -6: Chapter 7 I the questios below, describe each sequece recursively Iclude iitial coditios ad assume that the sequeces begi with a a = 5 As: a = 5a,a = 5 The Fiboacci umbers As: a = a + a, a = a = 3 0,,0,,0,, As: a = a, a = 0, a = 4 a = As: a = a +, a = 5 3,,,0,,, As: a = a, a = 3 6 a =! As: a = a, a = 7 /,/3,/4,/5, a As: a =, a = / + a 8 0, 0, 0, 0, As: a = a + /0, a = 0 9,,3 3,4, As: a = a +, a = 0,,,, As: a = 00a + a = the umber of subsets of a set of size As: a = a, a =,0,00,000, As: a = 00a +, a = Page 88

2 3 a = the umber of bit strigs of legth with a eve umber of 0s As: a = a +, a = 4 a = the umber of bit strigs of legth that begi with As: a = a, a = 5 a = the umber of bit strigs of legth that cotai a pair of cosecutive 0s As: a = a + a +, a = 0, a = 6 a = the umber of ways to go dow a -step staircase if you go dow,, or 3 steps at a time As: a = a + a + a 3, a = 0, a =, a 3 = 7 Verify that a = 6 is a solutio to the recurrece relatio a = 4a 3a As: = 6 = 6 8 Verify that a = 3 is a solutio to the recurrece relatio a = 4a 3a As: = = 3 3 = 3 9 Verify that a = is a solutio to the recurrece relatio a = 4a 3a As: = = = Verify that a = 3 + is a solutio to the recurrece relatio a = 4a 3a As: 4(3 + ) 3(3 + ) = = 3 (4 ) + = 3 + Verify that a = 7 3 π is a solutio to the recurrece relatio a = 4a 3a As: 4(7 3 π) 3(7 3 π) = π + 3π = 7 3 π Use the followig to aswer questios -6: I the questios below fid a recurrece relatio with iitial coditio(s) satisfied by the sequece Assume a 0 is the first term of the sequece a = As: a = a, a 0 = 3 a = + As: a = a, a 0 = 4 a = ( ) As: a = a, a 0 = 5 a = 3 As: a = a + 3, a 0 = Page 89

3 6 a = As: a = a, a 0 = 7 You take a job that pays $5,000 aually (a) How much do you ear years from ow if you receive a three percet raise each year? (b) How much do you ear years from ow if you receive a five percet raise each year? (c) How much do you ear years from ow if each year you receive a raise of $000 plus two percet of your previous year's salary As: (a) 5, (b) 5, (c) 5, , 000( ) 8 Suppose iflatio cotiues at three percet aually (That is, a item that costs $00 ow will cost $03 ext year) Let a = the value (that is, the purchasig power) of oe dollar after years (a) Fid a recurrece relatio for a (b) What is the value of $00 after 0 years? (c) What is the value of $00 after 80 years? (d) If iflatio were to cotiue at te percet aually, fid the value of $00 after 0 years (e) If iflatio were to cotiue at te percet aually, fid the value of $00 after 80 years As: (a) a = a /03 (b) a 0 = / (c) a 80 = / (d) / 0 05 (e) / Use the followig to aswer questios 9-34: I the questios below determie whether the recurrece relatio is a liear homogeeous recurrece relatio with costat coefficiets 9 a = 07a 03a As: Yes 30 a = a As: No 3 a a = 5 3a As: No 3 a = a 3 As: Yes 00 Page 90

4 33 a 7a + a 5 = 0 As: Yes 34 a + a = As: No 35 A vedig machie dispesig books of stamps accepts oly $ cois, $ bills, ad $ bills Let a deote the umber of ways of depositig dollars i the vedig machie, where the order i which the cois ad bills are deposited matters (a) Fid a recurrece relatio for a ad give the ecessary iitial coditio(s) (b) Fid a explicit formula for a by solvig the recurrece relatio i part (a) a = α + + β ) where As: (a) a = a + a, a 0 =, a = (b) ( ) ( α = ( + ) ad β = ( ) 36 Fid the solutio of the recurrece relatio a = 3a with a 0 = As: a = 3 Use the followig to aswer questios 37-45: I the questios below solve the recurrece relatio either by usig the characteristic equatio or by discoverig a patter formed by the terms 37 a = 5a 4a, a 0 =, a = 0 As: a = ( /3) 4 + (4/3) 38 a = 5a 4a, a 0 = 0, a = As: a = (/3) 4 (/3) 39 a = 0a a, a 0 =, a = As: a = ( 7/4)( 7) + (5/4) ( 3) 40 a = a, a 0 =, a = As: a = (/) + (3/) ( ) 4 a = a + a, a 0 = 0, a = As: ( 36)( + 3) ( 36 / )( 3) a = / 4 a = 3a, a 0 = As: a = 3! Page 9

5 43 a = a + 3, a 0 = 5 ( ) As: a = + 44 a = a + 5, a 0 = 3 As: a = 3 + 5( ) = a = a + +, a 0 = 5 As: a = 5 + ( + ) + = The solutios to a = 3a + 8a have the form a = c 3 + d ( 6) Which of the followig are solutios to the give recurrece relatio? (a) a = ( 6) (b) a = 5( 6) (c) a = 3c 6d (d) a = 3 (e) a = π(3 + ( 6) ) (f) a = 3 (g) a = 3 ( + ( ) ) (h) a = As: (a) Yes (b) Yes (c) No (d) Yes (e) Yes (f) Yes (g) Yes (h) No 47 Assume that the characteristic equatio for a homogeeous liear recurrece relatio with costat coefficiets is (r 5) 3 = 0 Describe the form for the geeral solutio to the recurrece relatio As: a = c5 + d5 + e 5 48 Assume that the characteristic equatio for a homogeeous liear recurrece relatio with costat coefficiets is (r + )(r + 4) = 0 Describe the form for the geeral solutio to the recurrece relatio As: a = c( ) + d( 4) + e( 4) 49 Assume that the characteristic equatio for a homogeeous liear recurrece relatio with costat coefficiets is (r + ) 4 (r ) 4 = 0 Describe the form for the geeral solutio to the recurrece relatio As: a = c( ) + d( ) + e ( ) + f 3 ( ) + g + h + i + j 3 50 Assume that the characteristic equatio for a homogeeous liear recurrece relatio with costat coefficiets is (r 3) (r 4) 3 (r + 7) = 0 Describe the form for the geeral solutio to the recurrece relatio As: a = c3 + d3 + e 3 + f4 + g4 + h 4 + i( 7) + j( 7) Page 9

6 5 The Catala umbers C also cout the umber of strigs of +'s ad 's with the followig property: as each strig is read from left to right, the umber of +'s ecoutered is always at least as large as the umber of 's (a) Verify this by listig these strigs of legths, 4, ad 6 ad showig that there are C, C, ad C 3 of these, respectively (b) Explai how coutig these strigs is the same as coutig the umber of ways to correctly parethesize strigs of variables As: (a) C : +, C : + +,+ +, C 3 : + + +, + + +, + + +, + + +, (b) Treat each + as a left parethesis ad each as a right parethesis 5 What form does a particular solutio of the liear ohomogeeous recurrece relatio a = 4a 4a + F() have whe F() =? As: p 0 53 What form does a particular solutio of the liear ohomogeeous recurrece relatio a = 4a 4a + F() have whe F() =? As: (p + p 0 ) 54 What form does a particular solutio of the liear ohomogeeous recurrece relatio a = 4a 4a + F() have whe F() = 4? As: (p + p + p 0 )4 55 What form does a particular solutio of the liear ohomogeeous recurrece relatio a = 4a 4a + F() have whe F() = ( + )? As: p + p+ p ( ) 0 56 Cosider the recurrece relatio a = a + 3 (a) Write the associated homogeeous recurrece relatio (b) Fid the geeral solutio to the associated homogeeous recurrece relatio (c) Fid a particular solutio to the give recurrece relatio (d) Write the geeral solutio to the give recurrece relatio (e) Fid the particular solutio to the give recurrece relatio whe a 0 = As: (a) a = a (b) a = c (c) a = 3 6 (d) a = c (e) a = Page 93

7 57 Cosider the recurrece relatio a = a + (a) Write the associated homogeeous recurrece relatio (b) Fid the geeral solutio to the associated homogeeous recurrece relatio (c) Fid a particular solutio to the give recurrece relatio (d) Write the geeral solutio to the give recurrece relatio (e) Fid the particular solutio to the give recurrece relatio whe a 0 = As: (a) a = a (b) a = c( ) (c) a = + 4 (d) a 4 ( ) = + + c (e) 3 a ( = ) 58 Cosider the recurrece relatio a = 3a + 5 (a) Write the associated homogeeous recurrece relatio (b) Fid the geeral golutio to the associated homogeeous recurrece relatio (c) Fid a particular solutio to the give recurrece relatio (d) Write the geeral solutio to the give recurrece relatio (e) Fid the particular solutio to the give recurrece relatio whe a 0 = As: (a) a = 3a (b) a = c3 a 5 3 = + + (c) a + 5 = (d) a + 5 = + c3 (e) 59 Cosider the recurrece relatio a = a + (a) Write the associated homogeeous recurrece relatio (b) Fid the geeral golutio to the associated homogeeous recurrece relatio (c) Fid a particular solutio to the give recurrece relatio (d) Write the geeral solutio to the give recurrece relatio (e) Fid the particular solutio to the give recurrece relatio whe a 0 = As: (a) a = a (b) a = c (c) a = (d) a = c (e) a = + 60 Suppose f () = 3 f (/) +, f () = Fid f (8) As: 40 6 Suppose f () = f (/3) +, f () = Fid f (7) As: 79 6 Suppose f () = f (/), f (8) = Fid f () As: /4 63 Suppose f () = f (/) + 3, f (6) = 5 Fid f () As: 5/4 64 Suppose f () = 4 f (/) + +, f () = Fid f (8) As: 6 Page 94

8 65 Use geeratig fuctios to solve a = 3a +, a 0 = 5 As: a = Use geeratig fuctios to solve a = 5a +, a 0 = + 5 As: a = 4 4 Use the followig to aswer questios 67-76: I the questios below write the first seve terms of the sequece determied by the geeratig fuctio 67 (x + 3) As: (a) 9,6,,0,0,0,0 68 ( + x) 5 As:,5,0,0,5,,0 69 ( + x) 9 As:,9,36,84,6,6, x As:,3,9,7,8,43,79 x x As: 0,0,,,,, + x x As:,,,,,, 73 5 As: 5,0,0,0,0,0,0 74 e x + e x As: 0,, 0,,, 0, cosx As: 0,,, 0,, 0,! 4! 6! Page 95

9 76 3 x x x As:,,0,0,,, Use the followig to aswer questios 77-87: I the questios below fid the coefficiet of x 8 i the power series of each of the fuctio 4 77 ( x x ) As: ( x x x ) As: ( x x x x ) As: ( x x x x x ) As: 5 8 ( + x 3 ) As: 0 8 ( x)( x )( x 3 )( x 4 )( x 5 ) As: 3 x As: x 3x As: ( x) As: 9 86 x (+ x) As: 7 6 Page 96

10 87 3x As: 3 4 Use the followig to aswer questios 88-00: I the questios below fid a closed form for the geeratig fuctio for the sequece 88 4,8,6,3,64, 4 As: x 89,0,,0,,0,,0, As: x 90,0,0,,0,0,,0,0,, As: 3 x 9,4,6,8,0,, As: ( x) 9 0,0,0,,,,,0,0,0,0,0,0,0,0,0, 3 3 x + x+ x + x As: ( ) 93,3,4,5,6,7, x As: + = x x x ( ) ( ) 94 0,,,0,,,0,,,0,,,0, As: 3 x x 95,,!, 3!, 4!, 5!, As: e x 96,!, 4!, 6!, 8! As: e Page 97

11 97,,,,,,,, As: + x 98,0,,0,,0,,0,,0,, As: + x ,,,,,, 000,,, As: ( + x) ,, 3,, 50, 0, 0, 0, 3 50 As: 50( + x) 49 0 Set up a geeratig fuctio ad use it to fid the umber of ways i which eleve idetical cois ca be put i three distict evelopes if each evelope has at least two cois i it As: (x + x 3 + x 4 + ) 3, 0 Set up a geeratig fuctio ad use it to fid the umber of ways i which eleve idetical cois ca be put i three distict evelopes if each evelope has most six cois i it As: ( + x + x + + x 6 ) 3, Set up a geeratig fuctio ad use it to fid the umber of ways i which eleve idetical cois ca be put i three distict evelopes if o evelope is empty As: (x + x + x 3 + ) 3, Set up a geeratig fuctio ad use it to fid the umber of ways i which eleve idetical cois ca be put i three distict evelopes if each evelope has a eve umber of cois i it As: ( + x + x 4 + x 6 + ) 3, 0 05 Set up a geeratig fuctio ad use it to fid the umber of ways i which eleve idetical cois ca be put i three distict evelopes if each evelope has at least two but o more tha five cois i it As: (x + x 3 + x 4 + x 5 ) 3, Page 98

12 06 Set up a geeratig fuctio ad use it to fid the umber of ways i which eleve idetical cois ca be put i three distict evelopes (labeled A, B, C) if evelope A has at least three cois i it As: (x 3 + x 4 + x 5 + x 6 + )( + x + x + x 3 + ), Set up a geeratig fuctio ad use it to fid the umber of ways i which eleve idetical cois ca be put i three distict evelopes (labeled A, B, C) evelopes A ad B have the same umber of cois i them As: ( + x + x 4 + x 6 + x 8 + x 0 )( + x + x + x 3 + ), 6 08 Set up a geeratig fuctio ad use it to fid the umber of ways i which ie idetical blocks ca be give to four childre if each child gets at least oe block As: (x + x + x 3 + ) 4, Set up a geeratig fuctio ad use it to fid the umber of ways i which ie idetical blocks ca be give to four childre, if each child gets at least two blocks As: (x + x 3 + x 4 + ) 4, 4 0 Set up a geeratig fuctio ad use it to fid the umber of ways i which ie idetical blocks ca be give to four childre, if each child gets at most five blocks As: ( + x + x + x 3 + x 4 + x 5 ) 4, 40 Set up a geeratig fuctio ad use it to fid the umber of ways i which ie idetical blocks ca be give to four childre, if the oldest child gets three blocks As: x 3 ( + x + x + x 3 + ) 3, 8 Set up a geeratig fuctio ad use it to fid the umber of ways i which ie idetical blocks ca be give to four childre, if the oldest child gets at most three blocks As: ( + x + x + x 3 )( + x + x + x 3 + ) 3, 64 3 Set up a geeratig fuctio ad use it to fid the umber of ways i which ie idetical blocks ca be give to four childre, if the oldest child gets either or 3 blocks As: (x + x 3 )( + x + x + x 3 + ) 3, 64 4 If G(x) is the geeratig fuctio for a 0,a,a,a 3,, describe i terms of G(x) the geeratig fuctio for 0,0,0,a 0,a,a, As: x 3 G(x) 5 If G(x) is the geeratig fuctio for a 0,a,a,a 3,, describe i terms of G(x) the geeratig fuctio for 0,0,0,a 3,a 4,a 5, As: G(x) a 0 a x a x Page 99

13 6 If G(x) is the geeratig fuctio for a 0,a,a,a 3,, describe i terms of G(x) the geeratig fuctio for a 3,a 4,a 5,a 6, As: ( Gx ( ) a0 ax ax ) 3 x 7 If G(x) is the geeratig fuctio for a 0,a,a,a 3,, describe i terms of G(x) the geeratig fuctio for a 0,0,a,0,a,0,a 3,0,a 4, As: G(x ) 8 If G(x) is the geeratig fuctio for a 0,a,a,a 3,, describe i terms of G(x) the geeratig fuctio for a 0,3a,9a,7a 3,8a 4, As: G(3x) 9 If G(x) is the geeratig fuctio for a 0,a,a,a 3,, describe i terms of G(x) the geeratig fuctio for a 0,0,0,a,0,0,a,0,0,a 3, As: G(x 3 ) 0 If G(x) is the geeratig fuctio for a 0,a,a,a 3,, describe i terms of G(x) the geeratig fuctio for 5,a,0,a 3,a 4,a 5, As: G(x) a 0 a x + 5 Use geeratig fuctios to solve a = 5a + 3, a 0 = 3 As: a = Use geeratig fuctios to solve a = 7a 0a, a 0 =, a = 4 As: a = Use geeratig fuctios to solve a = 3a + + 5, a 0 = + 5 As: a = 3 4 Fid A A A 3 A 4 if each set A i has 00 elemets, each itersectio of two sets has 60 elemets, each itersectio of three sets has 0 elemets, ad there are 0 elemets i all four sets As: 0 5 Fid A A A 3 A 4 if each set A i has 50 elemets, each itersectio of two sets has 80 elemets, each itersectio of three sets has 0 elemets, ad there are o elemets i all four sets As: 00 6 Fid the umber of terms i the formula for the umber of elemets i the uio of four sets give by the priciple of iclusio-exclusio As: 5 Page 00

14 7 Fid the umber of positive itegers 000 that are multiples of at least oe of 3,5, As: 55 8 Fid the umber of positive itegers 000 that are multiples of at least oe of,6, As: Fid the umber of positive itegers 000 that are multiples of at least oe of 3,4, As: Suppose A = B = C = 00, A B = 60, A C = 50, B C = 40, ad A B C = 75 How may elemets are i A B C? As: 5 3 How may positive itegers ot exceedig 000 are ot divisible by either 4 or 6? As: A doughut shop sells 0 kids of doughuts You wat to buy 30 doughuts How may possibilities are there if you wat at most six of ay oe kid? As: 33 A doughut shop sells 0 kids of doughuts You wat to buy 30 doughuts How may possibilities are there if you wat at most of ay oe kid? As: A market sells te kids of soda You wat to buy bottles How may possibilities are there? if you wat (a) at least oe of each kid (b) at most seve bottles of ay kid? As: (a) (b) A market sells te kids of soda You wat to buy bottles How may possibilities are there? if you wat at most three bottles of ay kid? As: 36 Suppose you have 00 idetical marbles ad five jars (labeled A, B, C, D, E) I how may ways ca you put the marbles i the jars if: (a) each jar has at least six marbles i it? (b) each jar has at most forty marbles i it? As: (a) 74 (b) 4 Page 0

15 37 How may ways are there to choose five douts if there are eight varieties ad oly the type of each dout matters? As: 7 38 A market sells 40 kids of cady bars You wat to buy 5 cady bars (a) How may possibilities are there? (b) How may possibilities are there if you wat at least three peaut butter bars ad at least five almod bars? (c) How may possibilities are there if you wat exactly three peaut butter bars ad exactly five almod bars? (d) How may possibilities are there if you wat at most four toffee bars ad at most six mit bars? As: (a) 54 (b) 46 (c) 44 (d) How may permutatios of all 6 letters of the alphabet are there that cotai at least oe of the words DOG, BIG, OIL? As: 4! 3 40 How may permutatios of all 6 letters of the alphabet are there that cotai at least oe of the words CART, SHOW, LIKE? As: 3 3! 3 0! + 7! 4 How may permutatios of all 6 letters of the alphabet are there that cotai at least oe of the words SWORD, PLANT, CARTS? As: 3! 8! 4 How may permutatios of all 6 letters of the alphabet are there that cotai oe of the words: SAVE, PLAY, SNOW? As: 6! 3 3! + 0! 43 How may permutatios of all 6 letters of the alphabet are there that cotai at least oe of the words: CAR, CARE, SCAR, SCARE? As: 4! 44 How may permutatios of the 6 letters of the alphabet are there that do ot cotai ay of the followig strigs: LOP, SLOP, SLOPE, LOPE As: 6! 4! 45 You have te cards, umbered through 0 I how may ways ca you put the te cards i a row so that card i is ot i spot i, for i =,,,0? As: D = 0! 0 9!+ 0 8! 0 7!++ 0 0! Page 0

16 46 Suppose A = 8 ad B = 4 Fid the umber of fuctios f : A B that are oto B 8 As: A office maager has four employees ad ie reports to be doe I how may ways ca the reports be assiged to the employees so that each employee has at least oe report to do 9 As: A office maager has five employees ad projects to be completed I how may ways ca the projects be assiged to the employees so that each employee works o at least oe project As: Fid the umber of ways to put eight differet books i five boxes, if o box is allowed to be empty 8 As: Fid the umber of bit strigs of legth eight that cotai a pair of cosecutive 0s As: a = a + a +, a = 0, a = Hece a 8 = 0 5 Fid the umber of ways to climb a -step staircase, if you go up either oe or three steps at a time As: a = a + a 3, a = a =, a 3 = Hece a = 60 5 Fid the umber of strigs of 0s, s, ad s of legth six that have o cosecutive 0s As: a = a + a, a = 3, a = 8 Hece, a 6 = 448 Page 03