12.1 Arithmetic Sequences & Series

Transcription

1 . Arithmetic Sequeces & Series. A sequece is ordered list of umbers. Ech umber i the list is clled term of the sequece. The first term of sequece is deoted s. The secod term is deoted s. The term i the th positio is clled the th term d is deoted s. The term before is. A sequece is fuctio whose rge is the terms of the sequece d the domi is the positio of ech term.. Fidig the Next Term: To fid the ext term i rithmetic sequece, first fid the commo differece, the dd the commo differece to the ext term.. Fid the ext three terms..,, 0,... b. 7, 0,, c. r 4, r, r + 4. Arithmetic sequeces: The sequece,,, 4,. is rithmetic if there is umber d such tht: = d = d Where d is the commo differece. Ex: 6, 9,,,. + The commo differece is becuse 9 6 =. d = Ex:, -, -8, -,, The commo differece is becuse - = -. d = -. Explicit Formul: formul tht defies the th term. = d + c We will look t c s beig the 0 ( sub ot) term. Thik y-itercept! The book will use d or d (whe recursive) 6. Recursive Sequece: is sequece i which ech term is defied usig the previous terms. Ech Arithmetic Sequece c be writte recursively usig Ex: Fid formul of rithmetic sequece whose commo differece is 4 d whose first term is. = d + c We kow d = 4. =. So 0 = 4. 0 = - = 4. The terms of this sequece re:, 7,,,, 4. Ex: Fid the formul of the rithmetic sequece whose first term is d whose secod term is. = d + c We kow = d = -. So d = must be (-4) = 7 = The terms of this sequece re:, -, -, -9,, d

2 7. Fid the th term i the sequece, 4, -, 8. Fid the 0 th term i the sequece for which = -7 d d =. 9. Fid the first term i the sequece for which 4 = 9 d d = The fifth term of rithmetic sequece is d the th term is 60. Write the first severl terms of this sequece. = = 60 = + 7d (where 7 is the differece i the term umbers). 60 = + 7d = 7d d = Sice = we c subtrct to get ech term i the sequece dow to the first., 0,, 0,. Arithmetic mes: the terms betwee y two ocosecutive terms of rithmetic sequece. The terms betwee give terms of rithmetic sequece re clled rithmetic mes. 0,, 6, 9, 0, 4, 8, rithmetic mes rithmetic mes. Form rithmetic sequece tht hs five rithmetic mes betwee - d 9.. Form rithmetic sequece tht hs six rithmetic mes betwee - d.

3 4. Summtio Nottio: the sum of sequece is lso kow s Arithmetic Series. m k c k c c c... c m. Sigm Nottio: the sum of the first terms of sequece (clled series) Ex: i i 8 Ex: (k ) Ex: ( k 0 j j ) 6. The Sum of Arithmetic Series: S = This mes tht we dd the first d lst terms, the multiply by the umber of terms divided by. Ex: Fid the sum of the itegers from to 00. S = = 00, = d = 00 S = = 0(0) =, 0 7. Fid the sum of the first 7 terms i the series -4, -8, -, Fid the sum of the first terms i the series -, -6, 0,. 9. Fid for series for which =, d =, d S Nimish strts college svigs ccout for her dughter o her sixth birthdy. She pls to deposit $ the first moth d the icrese the deposit by $ ech moth. How much will she hve deposited i twelve yers?. The umber of sets i the first row is 0, the secod row is, the third row is 6, d so o. How my sets re i Row 6? How my sets is there ltogether i those 6 rows?

4 DAY HW. pg #-7 odd, 4, 4 Fid the ext five terms i ech rithmetic sequece.., 9,, 7., -, -7, 9..,, 4.,., 0,,. b, -b, -b,. Fid the 79 th term i the sequece -7, -4, -,. 7. Form rithmetic sequece tht hs oe rithmetic me betwee d. Solve. Assume tht ech sequece is rithmetic sequece. 9. Fid the 9 th term i the sequece for which = d d = -.. Fid for the sequece for which = 7, = -, d d =.. Fid the first term i the sequece for which d = - d 7 =.. Fid d for the sequece for which = 4 d = Fid the sixth term i the sequece,, Fid the 4 rd term i the sequece -9, -, -,. Form sequece tht hs oe rithmetic me betwee 6 d 48.. Form sequece tht hs two rithmetic mes betwee d 0.. Fid the sum of the first terms i the series Fid for series for which = -7, d =., d S = Terri works fter school t the Fid Foods Supermrket. Oe dy, Terri hd to stck cs of soup i grocery disply i the form of trigle. O the top row, there ws oly oe c. Ech row below it cotied oe more c tht the oe bove it. O the bottom row, there were cs. If ll the cs were the sme size, how my cs were i the disply? 4. Michel is chocoholic. O New Yer s Dy, he te oe piece of chocolte. O the ext dy, he te pieces. O ech subsequet dy, he te oe dditiol piece of cdy.. How my pieces of cdy did he et o the lst dy of Jury? b. How my pieces of did he et durig the moth of Jury?

5 Dy Sequeces d Series Notes. Recursive Formul: A formul for sequece tht gives the vlue of term i terms of the precedig term. The first term is represeted by, the secod term i represeted by, the third term i represeted by, d so forth. Explicit Formul: d or d if Arithmetic Sequece!. Fid the ext three terms i ech sequece.. 80, 77, 74, 7, 68, b. 4, 8, 6,, 64, *c. 0,, 7,, 8, * d.,,,, Now write the recursive formul for the sequeces bove. 4. If d, fid the ext three terms.. If t 64 d t t, fid the ext four terms. 6. If, d 4, fid the third, fourth d fifth terms.. 7. Fid the first four terms.. t 6 b. 4k k c.

6 8. If the domi vlues re,0,,, fid the correspodig rge vlues for t. 0. Sigm Nottio Simplifies the process of writig out the sum of series first vlue of 0 lst vlue of is red s the sum of s icreses from to 0. k k. k ( k ). k 0 k 4. (o pretheses, creful) k k k [ 4( )] (typo: must be k s)

7 Homework: Worksheet # d Worksheet # DAY HW Worksheet I. Give the first four terms of ech sequece:. t, t t. t 0, t t. t, t t 4. t, t t 4 II. Give the third, fourth, d fifth terms of ech sequece:. t 6, t t 4 6. t 9, t t 7. t, t t 8. t 4, t ( t ) 0 9. t, t t 0. t, t ( t ). t d t 4, t t t. t d t 4, t t t. t 4. t 7d t, t t t d t 8, t ( t t ) Worksheet I. The domi of the sequece i ech exercise cosists of the itegers,,,4,. Write the correspodig rge vlues:.. k. k II. Write the first four terms of the sequece give by the formul i ech exercise k 0 k 9. III. Fid the sum of the first five terms of the sequece give by the formul i ech exercise: Let = 7, fid the sum: IV. Evlute ech of the followig:

8 k 4.. ( ) k k k 6 6. [ ( j )] 7. j k 0 k. Geometric Sequeces d Series Notes Geometric Sequece: the rtio of y term to the previous term is costt. r = commo rtio Fid the ext three terms i the geometric sequece: Ex: 7,, 67, Ex:, 4, 8, 6 Ex: 8, 4, 6 Ex: = 4 Explicit Formul: formul tht defies the th term. r Ex: Fid the first terms of the geometric sequece whose first term is = 4 d whose rtio is r =. Ex: Write rule for the th term. -8, -, -8, -7, Ex: 4 =, r =. Write rule for the th term. Ex: Fid the 8 th term of the geometric sequece whose first term is 0 d whose commo rtio is.. Ex: Fid the th 7 term i the geometric sequece 4, 6, 9,... Geometric Mes: the terms betwee y two ocosecutive terms of geometric sequece.

9 Ex: Isert geometric mes betwee 8 d. 8,,, Ex: Form sequece tht hs two geometric mes betwee 6 d 49. Ex: Form sequece tht hs two geometric mes betwee 8 d 4.. Geometric series S r r = commo rtio = umber of terms r works for fiite sequece Ex: Fid the sum of the first 0 terms of the geometric series Ex: Fid the sum of the st 8 terms of the geometric series where = 8 d 4 =. Ex: Durig the first week of triig of mrtho, you ru totl of 0 miles? You icrese the distce you ru ech week by twety percet. How my miles do you ru durig your twelfth week of triig? Ex: A certi chemicl decys t the rte of oe-hlf its weight every 6 moths. If 00 grms is the iitil weight of the chemicl, how much remis fter 4 yers? Ex: Whe bll bouces, the heights of cosecutive bouces form geometric sequece. Suppose bll is dropped from 0 meters d rebouds 9% of the height of the previous bouce. Wht is the height o the fifth bouce? **Wht is the totl distce trveled i the first five bouces?

10 . Book problems pg #7- odd DAY HW 7. Fid the ext four terms for the followig geometric sequece: 7,., 9. Determie whether,, 8,... form geometric sequece. Write yes or o.. Determie whether t, t,,... form geometric sequece. Write yes or o.. If r = d = 4, fid the first term of the geometric sequece.. Fid the sum of the first five terms of the series The first term of geometric sequece is 8, d the commo rtio is, fid the ext three terms. 9. Fid the sixth term of the geometric sequece 0, 0., 0.00,. Fid the first three terms of the geometric sequece for which 4 =. d r =.. Form sequece tht hs two geometric mes betwee d 7.. Fid the sum of the first six terms of the series Fid the sum of the first te terms of the series The Ldbury Museum hs bee ivestig i pitigs for my yers. Twety yers go, the museum purchsed pitig by oe of the Frech impressioist piters for $80,000. The vlue of the pitigs hs pprecited t rte of 4% per yer. Fid the vlue of the pitig fter 0, 0, 0, 40, d 0 yers, ssumig tht the rte of pprecitio remis costt.

11 . The popultio of certi bcteri doubles every 0 miutes. The iitil popultio is 0. Determie the umber of orgisms tht would exist fter hours d fter 4 hours. Worksheet Dy 4: Geometric/Arithmetic Sequeces d Series: Fid the th term of the Arithmetic Sequece hvig the give vlues of, d, d.. 6, d, 0. x, d x, Fid the specified term of ech rithmetic sequece:. The twetieth term of x y, x9y, x7y, Which term of, 08, 0,... is 7? Isert the specified umber of rithmetic mes i ech cse:. Seve, betwee 6 d 0 6. two, betwee x + y d x 4y Fid the missig terms: 7. t, t 7 t 4 6, 8. t, t 9 t 8, 7 Solve. 9. A rocket fired verticlly trveled 6 meters durig the first secod d (s log s its egies provided thrust) trveled 4 meters frther durig ech followig secod th i the oe before. How fr did it trvel durig the th secod? 0. If employee hired t $680 yer is gurteed ul slry icreses of $40, i which yer of his employmet will he first er t lest $0,000 yer?. A pile of bricks hs bricks i the first row, i the secod row, 49 i the third row, d so o, d brick i the top row. How my bricks re i the 4 th row? Use the give dt to fid the sum of the rithmetic series.., 4, d. 7, 7, d Solve: 4. A lecture hll hs 0 sets i the frot row d two sets more i ech followig row th i the precedig oe. If there re rows, wht is the setig cpcity of the hll?

12 . Roger pid off debt to his fther i 8 moths by pyig $60 the st moth, $ the secod, $0 the third, d so o. How much ws the origil debt? Write the ext three terms of the give geometric sequece, the write expressio for the th term. 6., 6,,. 7. 9,, Fid the specified term of the geometric sequece described. 8. teth term if d r 9. Eighth term of 4,, 6, Isert the give umber of positive geometric mes d write the resultig geometric sequece. 0. Two, betwee d 4. Four, betwee 48 d / Solve:. Assumig o duplictio of cestors, how my gret, gret, gret grdprets did you hve?. A piece of rel estte bought yers go for $,600 icresed i vlue % ech yer sice the. Wht is it worth ow? 4. A cr trveled meters i the first secod fter the brkes were pplied d i ech secod fter tht trveled hlf s fr s it hd i the secod before. How fr did the cr trvel i the te secods fter the brkes were pplied? Fid the sum of the geometric series described.., r, , r., 40. Of the five qutities,, r,, d S, three re give. Fid the other two. Some exercises hve two sets of swers. 7. 8, r, S 76 8.,, S 4

13 . d.4 Ifiite Sequeces & Series. Ifiite Sequece: A sequece tht ever eds.. The sequece,,,,,..., hs ifiitely my terms; s icreses the vlue of the 4 terms decrese d get closer d closer to 0. Thik bout the grph of this fuctio. 0 is clled the Limit of the terms i this sequece d c be expressed s follows: lim 0 the limit of, s pproches ifiity equls zero.. Rules for Fidig Limits (Sme s horizotl symptote rules):. If the lrgest expoets re the sme i the umertor d deomitor, the limit is the rtio of the coefficiets of the terms cotiig the lrgest expoet. b. If lrgest expoet is i the umertor, there is o limit. c. If the lrgest expoet is i the deomitor, the limit is Fid ech limit:. lim b. lim 4 6 c. lim 4 d. lim 4 *e. lim *f. lim. Ifiite Series: A ifiite series is the idicted sum of the terms of ifiite sequece 6. Sum of Ifiite Series: The sum of ifiite geometric series for which r is give by S r

14 7. Fid the sum of ech ifiite series, or stte tht the sum does ot exist..... b c Coverget Series: Hs sum or limit ( r ) 9. Diverget Series: Does ot hve sum or limit, ( r ) d it diverges 0. Determie whether ech series is rithmetic or geometric, the determie if it is coverget or diverget..... b c d. k k f. (.4) k. Repetig Decimls s Frctio: You c use wht you kow bout ifiite series to write repetig decimls s frctios - - you must write the repetig deciml s ifiite geometric series. Write ech repetig deciml s frctio b c

15 Homework: pge 676 # 7-4; Worksheet Dy DAY HW.4 Ifiite Geometric Series Determie whether ech series is coverget (hs sum) or diverget. If the series is coverget, fid the sum Decide whether or ot the ifiite series hs sum. If it does, fid it. 8. i i 9. i ()(0.) i Usig the ew iformtio, covert the followig to frctio form: Solve ech word problem.. Geologists estimte tht the cotiets of Europe d North Americ re driftig prt t rte of verge of miles every millio yers, or bout 0.7 ich per yer. If the cotiets cotiue to drift prt t tht rte, how my iches will they drift i 0 yers?

16 Cotgious diseses c spred very quickly. Suppose five people re ill durig the first week of epidemic d tht ech perso who is ill spreds the disese to four people by the ed of the ext week. By the ed of the teth week of the epidemic, how my people hve bee ffected by the illess. Book pg 676 #7-4 DAY HW Evlute ech limit, or stte tht the limit does ot exist. 7. lim 8. 6 lim 9. lim 4 0. lim. lim. lim. 4 lim 4. lim 8 4

17 . Sigm Nottio & The th Term I. Write ech of the followig i expded form d fid the sum. 7. ( 4 ).. Express usig sigm ottio. 4. Express usig sigm ottio.. Not ll sequeces re rithmetic or geometric; Some importt sequeces re geerted by products of cosecutive itegers Fctoril: The product ( )( )... is clled fctoril d is symbolized by! 6. By Defiitio 0! 7. Evlute ech expressio:.! b. 0! c. 0!! 6! d.! 6!! e.! ( )! f. 9! 0! g.! 6! 6!! h. ( x )!( x )! ( x 4)!( x )!

18 . Use Pscl s Trigle to expd ech biomil..6 Pscl s Trigle. b. ( x y) ( x ) c. ( x y) 6 d. ( ) e. ( x y)

19 Homework pge 688 # - odd, -8, 4, 4; Worksheet Dy 6 o Fctorils d Pscl s DAY 6 HW. Book pg 688 #- odd Write ech expressio i expded form d fid the sum.. r ( r ). (4 b ) 8 b4 7. b ( b b) 6 9. ( ) 4. p ( p ). k 4( ) k Worksheet --Fctorils: Dy 6 Write ech product i fctoril ottio: Evlute ech expressio usig fctoril kowledge: 6!.! (6!) 8.!4! 4!! 9. 0! 8! 0. 8! 4!.!!. 6! 7! 6!. 4! 6! 7! 4.! 0!.! 4!! 6.!!!!! 7.! 8.!! 9.!! 0. x y! x y!. x! x! x! x!. x! x! x! x! Expd usig Pscl s Trigle:. x y 6. b 4. x y 4 b. x

20 .6 Biomil Theorem. Biomil Theorem: If is oegtive iteger the b b b b b b ! 0!!...!!!!!! 0!!! 0 b b b k k k b k k 0!!!. Use the biomil theorem to expd ech expressio.. 4 x. Fid the idicted term of ech expressio.. Fourth term of 7 x b. Sixth term of 9 y x c. Fifth term of 9 y x d. Third term of 6 e. Fid the middle term of 9 4 y x r r r y x C y x x y x... where! )! (! r r C r if you re lookig for the fifth term the r = 4

21 4. Fid the term cotiig x y 8 y i the expsio of 9 Homework Worksheet o Specific Term Dy 7 DAY 7 HW.6 Specific Term Use the biomil theorem to fid the idicted term of the followig:. ( b ) 7 ; th term. (x + y) 7 ; 4 th term. (v + w) 0 ; 8 th term 4. (x + y) 0 ; rd term. ( x + y ) ; th term 6. (x y) 0 ; 4 th term 7. ( b ) ; 4 th term 8. x ; 8 th term 9. x x y ; 7 th term 0. x y x 0 ; 4 th term. Fid the middle term of x y. Fid the term which does ot coti x i the expsio of 0 6x. x. Fid the term cotiig y 6 i the expsio of (x y ) Fid the term cotiig c i the expsio of 0 c d.. Fid the term cotiig x i the expsio of (x + y) Fid the term cotiig y 0 i the expsio of x y ) Fid the middle term i the expsio of 0 b. 8. Fid the lst three terms i the expsio of ( b ) 7.

22 Aswers:. 80 b x 4 y. 40v w x 8 y. 00 x 0 y x 4 y b /x x 6 /y x /y x y x y c d. 4884x y 6. 79x y b / b +448 b 8 Sequeces d Series Word Problems Notes. The wier of cotest received $00 t the ed of the first yer, with % icrese over the precedig yer s pymet for ech subsequet yer. How much did the cotest wier receive durig the first 0 yers of pymets?. Mr. Turtle is movig log stright lie d hs trveled 8 meters i oe miute. I the ext miute he trvels 4 m. I ech succeedig miute he trvels ½ s fr s he did i the previous miute. If Mr. Turtle goes o trvelig this wy forever, how fr will he go?. The owers of certi store reduces the price of their items ech week the item does ot sell. If the origil price of blouse is $0 d its price t the ed of ech week is 4/ of the previous week, wht will be the price of the blouse t the ed of the 0 th week? 4. A pile of logs hs log i the top lyer, logs i the secod lyer, logs i the third lyer, d so o. How my logs re i the pile if it cotis lyers?. A frmer gthers bushels of pottoes o the first dy of the hrvest. The frmer estimtes tht o ech successive dy of the hrvest, the mout gthered will be 4 bushels more th the precedig dy. If the hrvest lsts 4 dys, wht is the totl umber of bushels the frmer c expect to collect? 6. Frictio d ir resistce cuse ech swig (fter the first) of the pedulum bob to be 7% s log s the previous swig. If the legth of the first swig is 6 cm, fid the totl distce trveled by the bob before comig to rest. Write ech i sigm ottio

23 DAY 8 HW Sequeces d Series Word Problems. Ech swig of pedulum is i. shorter th the precedig swig. The first swig is 6 ft. Write rule for the legth of ech swig i iches. How log is the th swig?. Oe yer before trip you begi mkig deposits to ccout. The first moth you deposit $40. For the ext moths you deposit $ more th the previous moth. Write rule for ech mothly deposit. How much moey do you hve fter moths?. You buy ew cr for $,000. The vlue of the cr decreses by 6% ech yer. Write rule for the verge yerly vlue of the cr (i dollrs) i terms of the yer. Let = curret yer. About how much will the cr be worth fter 6 yers? 4. A teis bll dropped from height of feet bouces 70% of the height from which it fell o ech bouce. Wht is the verticl distce it trvels before comig to rest?. Emily d Kyl begi work d receive $000 ech the first moth. Emily will receive rise of $0 ech moth therefter. Kyl will receive % rise ech moth therefter.. At the ed of moths, how much will Emily d Kyl ech be mkig per moth? Who hs the higher mothly icome? b. How much did Emily d Kyl mke durig the moth period? Who ered the higher yerly icome? 6. Kevi drops Super Boucer bll from his blcoy, 0 feet bove the groud. If the bll rebouds 90% of the height from which it fell o ech bouce, fid the verticl distce tht the bll trvels before comig to rest.

24 7. After oe miute, hot ir blloo rose 90 feet. After tht time, ech succeedig miute the blloo rose 70% s fr s it did the previous miute. How fr bove the Erth ws the blloo fter 8 miutes? Wht ws the mximum height of the blloo? 8. To prove tht objects of differet weights fll t the sme rte, Glileo dropped two objects with differet weights from the Leig Tower of Pis i Itly. The objects hit the groud t the sme time. Whe object is dropped from tll buildig, it flls bout 6 feet the first secod, 48 feet i the secod secod, d 80 feet i the third secod, regrdless of its weight. How my feet would object fll i the teth secod? 9. A costructio compy will be fied for ech dy it is lte completig its curret project. The dily fie will be $4000 for the first dy d will icrese $000 ech dy. Bsed o its budget, the compy c oly fford $60,000 i totl fies. Wht is the mximum umber of dys it c be lte? 0. The Uited Sttes Deprtmet of Defese pls to cut the budget o oe of its projects by % ech yer. If the curret budget is $0 millio, wht will the budget be i 6 yer?. A oe-to ice sculpture is meltig so tht it loses oe-fifth of its weight per hour. How much of the sculpture will be left fter five hours? Write the swer i pouds. ( to=000 lbs). Rob is helpig his dd istll fece. He is usig sledgehmmer to drive the poited posts ito the groud. O his first swig, he drives post iches ito the groud. Sice the soil is deser the deeper he drives, o ech swig fter the first, he c oly drive the post 0% s fr ito the groud s he did o the previous swig. How fr hs he drive the post ito the groud fter swigs?. A chessbord hs 64 squres. If oe pey is plced o the first squre, the doubled to two peies o the secod, the doubled to four peies o the third, how much moey will be o the bord whe the d squre is reched? 4. A bll is dropped from height of m d bouces to 60% of the previous height. How fr hs the bll trveled whe it hits the groud for the fourth time?

25 Write ech i sigm ottio ( ) TEST REVIEW. Which term of the rithmetic sequece 8,, 4, is 7?. Isert three rithmetic mes betwee d.. Fid the eleveth term of the rithmetic sequece 8,,, Give 4 d 8, fid.. Fid the sum of the first 7 terms i the rithmetic series Write usig sigm ottio. k 7. Fid ( ) k 8. Fid the sum of the rithmetic series with 70, 7, d 9. Fid the fifth term of the geometric sequece 4, -6, 4, 0. Isert four geometric mes betwee 48 d.. Fid the eighth term i the geometric sequece 4,, 6,. Which term of the geometric sequece 0, -0, 0, is 60?. Evlute ( ) k k 4. Fid the sum of the geometric series 48, r, d.. I the geometric series, give 8, r, S 76, fid d Fid the sum of the ifiite geometric series 6, Expd ( x y ) 4 usig Pscl. 8. Expd ( b) 9. Fid the fourth term i the expsio ( p q) 0. Fid the term i the expsio of. Evlute ech.. 7! b. 00! 99! 0 ( x y) cotiig c. 9! 6!! 9 x.. Expd ( b) usig Pscl. 4. Fid the sum of the ifiite series Write the first four terms i the sequece formed by the ptter t.. Fid the limit lim d. ( )!! Solutios. 4 th. 70, 8, (k 4) k ,, 6,. 6. th ; x x y 4x y 8xy y 4 b 0 b 0 b 8. 4 b b p q 0. 4 th 9 term 0x y b. 00 c. 84 d b 70 b b b b. 0 4.,, , 4 7. Does Not Exist , 7,,, 6 0. t 9, t t 4 8

26 8 6. Fid the limit lim 7 7. Fid the limit lim 8. Chge to frctio i lowest terms: Give the sequece t, t t, list the first five terms i the sequece. 0. Give recursive defiitio 9,, 7,,