Sequences and Series

Transcription

1 Secto 9. Sequeces d Seres You c thk of sequece s fucto whose dom s the set of postve tegers. f ( ), f (), f (),... f ( ),... Defto of Sequece A fte sequece s fucto whose dom s the set of postve tegers. The fucto vlues,,, 4,... re the terms of the sequece. If the dom of the fucto cossts of the frst postve tegers oly, the sequece s fte sequece.. Exmple: Wrte the frst four terms of the followg sequeces. ),... = () + = 7 = () + = 9 = () + = 4 = (4) + =

2 Secto 9. b) b b = - = 0 = b = - = = b = - = = 9 b 4 = 4- = = 7 c) c ( ) c c c c c 4 ( ) ( ) ( ) 4 ( ) 4 ( ) Notce how the sgs lterte.

3 Secto 9. Becuse dfferet sequeces c hve the frst terms the sme, t s ecessry to kow the th term order to defe uque sequece. The th term c be thought of s the rule for the sequece. Fdg the th Term of Sequece To fd the ppret ptter, lst the terms d udereth lst the umbers for. Look for ptter tht shows wht s doe to to get the term the ptter. Exmple: Wrte expresso for the ppret th term ( ) of the sequece,,, 7, : 4 Terms: 7 Look t how the term c be rrved t usg the gve vlue. It ppers tht we c double the vlue d the subtrct. Ths tells us tht = -.

4 Secto 9. Exmple: Wrte expresso for the ppret th term ( ) of the sequece, -4, 6, -8, 0, Soluto: = (-) + () : 4 Terms: Recursve Sequeces To defe sequece recursvely, you eed to be gve oe or more of the frst few terms. All other terms of the sequece re the defed usg prevous terms. Exmple:,,,,, 8,,, 4,. To get the ext term, we dd the prevous terms. But order to defe the sequece, we hve to lst the frst terms, becuse there re o terms before those terms tht we c dd. The sequece s defed s: =, =, = Ths s well kow sequece clled the Fbocc Sequece. 4

5 Secto 9. Fctorl Notto Whe we wt to multply product such s 4 =0 we use fctorl otto. Defto: If s postve teger, fctorl s defed s! = 4 (-) As specl cse, 0! =. Exmple: Evlute 6! Soluto: 6! = 6 4 = 70 Exmple: Wrte the frst 4 terms of the sequece defed by!,,!! 9,! ! 6 4

6 Secto 9. Note:! = ( 4 ) ()! = 4 Evlutg Fctorl Expressos Exmple: Evlute the fctorl expressos. ) 0!! 8! Expdg, we get: ()( ) After ccelg, we ed up wth: 4 You should be ble to go from 0!! 8! to 0 9. b)! 6!!!! 6!! 6 6 soluto:!!! 6

7 Secto 9. c)! ( )! soluto:! ( )! (! )! Fctorls Usg Grphg Clcultor The fctorl key fucto c be ccessed by [MATH] [PRB] [! ]. Exmple: Fd! usg your clcultor. Press [MATH] [PRB] [! ] [ENTER]. Soluto: The scree wll show the swer.. x0 Exmple: Fd 7!-6! usg your clcultor. Press 7 [MATH] [PRB] [! ] - 6 [MATH] [PRB] [! ] [ENTER]. Soluto: 40 7

8 Secto 9. Grphg Sequeces o Grphg Clcultor. Put the clcultor Sequece mode by pressg [MODE] d the o the 4 th le, hghlght Seq. Press [ENTER] d the [ d ] [QUIT].. Eter the sequece the [Y=] scree. Your sequece c be med u, v, or w. We wll use u for these structos. Set the M = to show tht your vlues for wll strt wth the umber. Type the formul for your th term fter u() =. (You c use the [X,T,θ,] key for.) You do ot hve to eter the frst term, u(m) uless your sequece s recursve.. Move your cursor to the very left of the u() =. By repetedly pressg [ENTER] you c choose how the grph wll be dsplyed. Select the dotted opto. 4. Press [GRAPH]. The press [ZOOM] [ZoomFt] for the best vewg wdow. Note: You c lso chge the vewg wdow from the [WINDOW] scree. You wll see tht you c set the m d mx for, s well s for x d y. 8

9 Exmple: Grph the frst 0 terms of the sequece 4 defed by u Exmple: Grph the frst 0 terms of the sequece defed by v ( ) () CHAT Pre-Clculus Secto 9. To Vew Idetfy Idvdul Terms Usg Grphg Clcultor:. Usg the [TRACE] feture: Press [TRACE] d the the UP ARROW oce. You should see dsplyed o your scree the vlues of d x (whch re detcl) d y (whch s the vlue of the term). Use the rght d left rrows to move from term to term.. Usg the [TABLE] feture: Press [TBLSET]. Set the followg vlues: TblStrt = (sce strts t ). Tbl = (sce the vlues go up by s). Idpt: Auto Deped: Auto Press [TABLE] to see the terms lsted. 9

10 Secto 9. Fdg the th Term Usg Grphg Clcultor Your clcultor hs key for ech of the sequeces u, v, d w. The keys re the [ d ] fucto of the 7, 8, d 9 keys respectvely. To fd the th term of sequece, do the followg:. Eter the sequece t the [Y=] scree.. Press [[ d ] [QUIT].. Press u (usg [ d ] [7]). The u should pper o your scree. 4. Type the umber of the term you wt, eclosed by pretheses d press [ENTER]. Exmple: Use Fd the th term of the sequece 4 u usg the followg methods:. Drect clculto of the term.. The [TRACE] feture o the grph of the sequece.. The [TABLE] feture. 4. Eterg u() drectly to the clcultor. Soluto: u. 0

11 Secto 9. Summto Notto We ofte wt to fd the sum of the terms of fte sequece. The otto we use s clled summto otto or sgm otto becuse t volves the Greek letter sgm, wrtte s. Defto of Summto Notto The sum of the frst terms of sequece s represeted by 4... where s clled the dex of summto, s the upper lmt of summto, d s the lower lmt of summto. Exmple: 4 4() 4() 4() 4(4) 4() 60

12 Secto 9. Exmple: Fd ech sum. 4 ) b) ( ) ( )!! ( ) 0 (!) ( ) (!) ( ) ( ) 6 ( 4) 0 0 (!) ( ) (4!) ( ) 4 (!) c) k ( k ) k ( k ) *Note: The lower lmt does ot hve to be d the dex does ot hve to be.

13 Secto 9. Propertes of Sums. costt. c s c c,. costt. c s c c,. b b ) ( 4. b b ) ( Seres My pplctos volve the sum of the terms of fte or fte sequece. Such sum s clled seres.

14 Secto 9. Defto of Seres Cosder the fte sequece,,,...,..., 4. The sum of the frst terms of the sequece s clled fte seres or the th prtl sum of the sequece d s deoted by The sum of ll terms of the fte sequece s clled fte seres d s deoted by Exmple: Cosder the seres 0. ) Fd the rd prtl sum

15 Secto 9. b) Fd the sum of the fte seres , Fdg Prtl Sums o Grphg Clcultor You c use the [sum] feture log wth the [seq(] feture to fd prtl sums of sequece. The fucto [sum( ] s used to fd the prtl sum of sequece. The formt for the [seq(] commd re: seq(expresso, vrble, beg, ed, cremet) (The defult for cremet s, so f your cremet s, you do ot hve to type t.)

16 To fd the kth prtl sum of the sequece, do the followg: Press [ d ][LIST] [MATH][sum( ] [ d ][LIST] [OPS][seq( ],,, k)) [ENTER]. CHAT Pre-Clculus Secto 9. If you hve the sequece lredy etered t the [Y=] scree, the you c do the followg: Press [ d ][LIST] [MATH][sum( ] [ d ][LIST] [OPS][seq( ] u,,, k)) [ENTER]. If you wt lst of the frst k prtl sums, do the followg: Press [ d ][LIST] [OPS][cumSum( ] [ d ][LIST] [OPS][seq( ],,, k)) [ENTER]. The frst k prtl sums wll be lsted s set { }. If you hve the sequece lredy etered t the [Y=] scree, the you c do the followg: Press [ d ][LIST] [OPS][cumSum( ] [ d ][LIST] [OPS][seq( ] u,,, k)) [ENTER]. 6

17 Secto 9. Exmple: Fd the 6 th prtl sum of the sequece defed by =. Soluto: Press [ d ][LIST] [MATH][sum( ] [ d ][LIST] [OPS][seq( ],,, 6)) [ENTER]. swer = 6 4 Exmple: Fd the sum by frst eterg the 0 sequece to the [Y=] scree. Soluto: After eterg the sequece to the sequece u, press Press [ d ][LIST] [MATH][sum( ] [ d ][LIST] [OPS][seq( ] u,,, 4)) [ENTER]. swer = 0. Exmple: Fd the frst prtl sums of the seres Soluto: Press [ d ][LIST] [OPS][cumSum( ] [ d ][LIST] [OPS][seq( ],,, )) [ENTER]. swer = {, 6, 4, 0, 6} 7

18 Secto 9. Applcto Exmple: If depost of $0 s mde ech moth to ccout tht ers 6% terest compouded mothly, the the blce the ccout fter moths s A = 0(00)[(.00) ]. Fd the blce the ccout fter yers. Soluto: yers s 60 moths, so = 60. Fd A 60. A 60 = 0(00)[(.00) 60 ] = $