A function f whose domain is the set of positive integers is called a sequence. The values


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1 EQUENCE: A fuctio f whose domi is the set of positive itegers is clled sequece The vlues f ( ), f (), f (),, f (), re clled the terms of the sequece; f() is the first term, f() is the secod term, f() is the third term,, f () is the th term, d so o I tretig sequeces, it is customry to use subscript ottio isted of fuctiol ottio Thus, for give sequece f, f () is deoted by, f () by, f () by,, d f () by I this ottio, we represet the sequece s,,,,, Exmples : Write the first four terms, the ith term d the twetieth term of the sequece whose th term is: () () () ) ( () ( ) olutios: To fid the first four terms we substitute, successively,,,, i the formul for The ith d twetieth terms re foud by substitutig 9 d 0 i the formul for The results re: First four terms Nith term Twetieth term (),,, (),, 5, (),, (),, 9,
2 Rther th givig explicit formul for, sequeces re sometimes specified by sttig method for fidig the terms Oe such method is clled recursio formul or recurrece reltio A recursio formul is formul tht gives i terms of oe or more of the terms tht precede The strtig vlue, or vlues, must lso be give We illustrte with some exmples Exmples : Fid the first four terms d the th term for the sequece specified by: () d, k,,, () d ( ), k,,, olutios: () We re give From the recursio formul,, Bsed o this ptter, we coclude tht 5, 5 6, d, i geerl () Proceedig s i (),,,, Bsed o this ptter, we coclude tht 5 5 0, , d, i geerl, ( )( )! * Exmple : The sequece defied recursively by: d * The product of the first turl umbers is clled fctoril d is deoted by!;!,!,!,!, etc
3 is kow s the Fibocci sequece The terms of the sequece re clled the Fibocci umbers The first ie Fibocci umbers re It c be show tht,,,, 5, 8,,, ( 5) ( 5) 5 The Fibocci umbers pper i wide vriety of pplictios rgig from the biologicl scieces to rt d rchitecture The sequece of rtios of cosecutive Fibocci umbers pproches the umber 5 which is the soclled golde rtio used by the Greeks thousds of yers go Here is other exmple of sequece defied by method for fidig the th term Exmple 5: List the first six terms of the sequece whose th term is the th prime umber olutio:,, 5, 7, 5, 6 It is iterestig to ote tht there is o kow formul for However, this is perfectly well defied sequece sice is defied for ech positive iteger I tretmets of sequeces it is commo to mke sttemets such s cosider the sequece whose first four terms re,,,, It is left to the reder: () to discover the ptter estblished by the terms, d () to determie formul for the geerl term I this cse, the ptter ppers to suggest tht 5 / 5, 6 / 6, d, i geerl, / However, cosider the sequece defied by
4 9 ( )( )( )( ) 5 As you c verify,, /, /, / d 5 9 The poit of this exmple is: listig the first few terms of sequece does ot, i fct, determie specific sequece To determie specific sequece you must be give formul for the th term or you must be give some method for determiig the th term (eg, recursio formul) ttemets such s determie the sequece whose first four terms re,,, must be qulified by sttemet such s ssume tht ptter cotiues s idicted Exmples 6: () The first four terms of sequece re give () ssume tht the ptter cotiues s idicted d fid the geerl formul for ; d (b) fid formul for i which / 5 π,, 5, 6 7, 8 () The first five terms of sequece re give () ssume tht the ptter cotiues s idicted d fid the geerl formul for ; d (b) fid formul for i which ,,,,, 5 olutios: () () (b) π ( )( )( )( ) 8 () () () ( )
5 (b) 76 7 ( ) ( )( )( )( )( 5) 0 70 Limits of sequeces: Let be give sequece We re ofte iterested i the behvior of the terms s gets lrger d lrger, symbolized by For exmple, cosider the sequece / ice <, < for ll ; the terms re decresig i vlue Also, if is lrge (sy,000,000, or 0,000,000), / is close to 0 Thus, we coclude tht the terms of the sequece / re decresig d tedig to 0 Here is grph of the sequece If we simply plot the vlues / o the rel lie, we get 0 d we c see tht the terms re heded towrd 0; they ever pss zero becuse / > 0 for every positive iteger Bsed o this behvior, we sy tht the limit of / s teds to ifiity is 0 This is symbolized by lim 0 or by 0 s I geerl, if the terms of the sequece pproch umber L s, the L is clled the limit of the sequece As suggested i the exmple bove, this is symbolized by
6 lim L or by L s Exmples 7: Determie whether the give sequece hs limit s () () () ) ( () ( ) olutios: () We write out the first severl terms of the sequece to get ide of its behvior: 5 6,,,,, 5 It ppers tht the terms re gettig closer d closer to 5 6 We c justify this coclusio by otig tht For lrge, is close to sice / is close to 0; s () As i (), we write out the first severl terms 5 0,,,,,,
7 It ppers tht the deomitors of these frctios re growig much fster th the umertors d so we guess tht 0 s This coclusio is justified by writig d otig tht / 0 d / 0 s Therefore 0 s () Writig out the first severl terms, we hve,,,,, The terms simply oscillte betwee d ; the terms oe umber L so the limit does ot exist re ot gettig close to () The first severl terms of this sequece re,, 9, 6, 5, 6, The terms get rbitrrily lrge i bsolute vlue d oscillte betwee positive d egtive; the sequece does ot hve limit
8 Two specil sequeces: We ll coclude this sectio o sequeces by cosiderig two specil types of sequeces Arithmetic sequeces: A rithmetic sequece is sequece i which the differece of successive terms is costt d Tht is, sequece is rithmetic sequece if d for every positive iteger The umber d is clled the commo differece for the sequece Note tht rithmetic sequece is defied by recursio formul Arithmetic sequeces re lso clled rithmetic progressios Exmples 8: Determie whether the give sequece is rithmetic If it is, give the commo differece (), 5, 8,,,, (),, 9, 6,,, (), 8,,, 6, olutios: () [ ] ( ) The sequece is rithmetic; d ; 9 5 The sequece is ot rithmetic () () [ ] ( ) The sequece is rithmetic with d uppose is rithmetic sequece with first term differece d We will use the recursio formul to fid formul for defiitio, we kow tht d Therefore 5 d commo From the d, d d, d d, d d, d we coclude tht
9 ( ) d Exmple 9: Fid the twelfth term of the rithmetic sequece whose first three terms re, 5, 9, olutio: d d Therefore, ( ) d () 5 Geometric sequeces: A geometric sequece, or geometric progressio, is sequece i which the rtio of successive terms is ozero costt r Tht is, is geometric sequece if d oly if r for every positive iteger The umber r is clled the commo rtio Note tht geometric sequece is defied by recursio formul: r Exmples 0: () The sequece 8,,,,, is geometric sequece Fid the commo rtio d give the fifth term of the sequece () The sequece 5,,,, is geometric sequece Fid the commo rtio 8 d give the sixth term olutios: () The commo rtio is r The fifth term is 5 8 5/ () The commo rtio is r The sixth term is the fifth term times 5 / ; the fifth term is ( / )( 5/8) 5/6 Therefore, the sixth term is 5 /
10 uppose tht is geometric sequece with first term We ll use the recursio formul to determie expressio for the geerl term : r, r r, r r, r, 5 r Bsed o this ptter, we coclude tht r Exmple : Fid the seveth term of the geometric sequece whose first three terms re,,, 8 olutio: The first term d the commo rtio r is / Therefore, by the formul derived bove, the seveth term is: 6 7 ( / ) 6 08 ERIE Let,,,,, be give sequece uppose we re sked to dd up ll the terms of the sequece Tht is, suppose we re sked to clculte ice we c oly dd fiite umber of umbers, it would pper tht ddig up ll the terms of sequece is impossible tsk However, there re istces where we c ssig vlue, umber, to ifiite sum We ll explore tht possibility here A sum of the form, where is give sequece, is clled series (lso clled ifiite series) ice we c dd up y fiite collectio of umbers, we ll form ew sequece by ddig up terms of the give sequece i systemtic mer
11 Let This ew sequece is clled the sequece of prtil sums of the give sequece The questios we wt to ddress re: () C we fid formul for the sequece of prtil sums derived from some give sequece? () If the swer to () is yes, does hve limit, sy, s? If the swers to () d () re yes, the we sy tht the sum of ll the terms of the sequece is ; we ve dded up ifiitely my umbers d gotte umber! Here re some exmples Exmples : Let be the sequece defied by ( ) This is the sequece The sequece of prtil sums is: 0 0,,,, Bsed o these results, we coclude tht whe is odd d 0 whe is eve The sequece c be represeted by whe is odd ( ) or by 0 whe is eve
12 Does hve limit s? Let be the sequece defied by ) ( The first three terms of the sequece of prtil sums is: Wht is for every positive iteger? Does hve limit s? Let be the sequece defied by This is the sequece of positive itegers,,,,,, The sequece of prtil sums is: 0 6 Wht is for every positive iteger? Does hve limit s? olutios: does ot hve limit; the sequece oscilltes betwee 0 d s ; The ifiite series hs fiite sum: The sequece,,,,, is rithmetic sequece with d d
13 ( ) Addig these two equtios, we get ( ) ( ) ( ) ( ) ( ) [ terms] olvig for, we get the result ( ) ( ) This sys tht the sum of the first positive itegers is The sequece of prtil sums does ot hve limit; s Arithmetic d geometric series: Let,,, be rithmetic sequece, with first term d commo differece d The, d, d, d, d, [ d] [ d] [ ( ) d] is rithmetic series The sequece of prtil sums is: ( d) d ( d) ( d) ( ) d ( d) ( d) ( d) ( ) d [ ( )] d Usig our result i () bove, formul) d ( ) ( ) (replce by i the
14 ( ) d or [ ( ) d] This is formul for the sequece of prtil sums of rithmetic series ice ( ) d, it follows tht ( ) d [ ( ) d] Therefore, c lso be writte s ( ) Now suppose tht, d commo rtio r The d,,, is geometric sequece with first term, r, r, r r r r, r is geometric series The sequece of prtil sums is: r r r r r r r Note first tht if r, the Now ssume tht r The r r r ( s) d r r r r r r r r r r
15 ubtrctig the secod equtio from the first gives r r d ( r) r Therefore, r provided r r This is formul for the sequece of prtil sums of geometric series Let s look t the behvior of s We c rewrite our formul for s r r r It is possible to show tht if < r <, the r 0 s, d it follows tht r 0 d r r Therefore, if the commo rtio r of geometric series stisfies < r <, (ie, r < ), the we defie the (fiite) sum of the ifiite geometric series r r r r to be, r < r Exmples : Fid the sum of the first 8 terms of the rithmetic sequece, 7,, 5, Fid the sum of the first 6 terms of the geometric sequece,,,, 8 Does the ifiite series 8
16 hve fiite sum? olutios: d d Usig the first rithmetic series formul, we hve 8 8 [() (8 )] (6 8) 6 Usig the secod formul, (8 ) d d r Therefore [ ] () ice r <, the ifiite geometric series hs the fiite sum r 6 Exercises: Fid the first five terms d the eighth term of the followig sequeces 0
17 ( ) (0) 8 9 ( ) ( ) 0 Fid the first five terms of the sequece defied by the give recursio formul If possible, fid formul for,,,, 5, 6, Determie whether the give sequece hs limit If it does, give the limit 7 8
18 9 0 () Determie whether the idicted sequece c be the first three terms of rithmetic or geometric sequece, d, if so, fid the commo differece or commo rtio d the geerl term, 6,,,,9, 5,,8, 6 7,65,6, 7,,, 6 8 8,,, 9,,, 9 0 7,, 8, Fid the teth term d the th term of the give rithmetic sequece, 6, 0,,, 9, 7,
19 , 7,,, 7, 65, 6, Let,,, be rithmetic sequece Fid the idicted qutities 5, d ; fid d , d ; fid d 0 0 7, d 0; fid d /, d / ; fid d Fid the teth term d the th term of the give geometric sequece 9 8,,,, 0, 6, 8,,, 06,, 6, 9, 5, Let,,, be geometric sequece Fid the idicted qutities, r ; fid d 6 6, r / ; fid d 5, r / ; fid d 6, r 05; fid d Determie whether the geometric series hs fiite sum If it does, fid it
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