Section 3.3: Geometric Sequences and Series


 Michael Preston
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1 ectio 3.3: Geometic equeces d eies Geometic equeces Let s stt out with defiitio: geometic sequece: sequece i which the ext tem is foud by multiplyig the pevious tem by costt (the commo tio ) Hee e some exmples of geometic sequeces: ) 9, 8, 36, 7, b), 8, 7, 8, c) 0, 30, 90, 70, d) 3,, 48, 9, e) 48, 36, 7, The commo tios of ech of these sequeces, i ode fom ) to e), is, 3, 3, 4, 3, 4 espectively. Note tht i ech of them, we c fid the commo tio by tkig y tem d dividig it by the pevious tem. Like y othe sequeces, geometic sequeces c be fiite o ifiite. c) bove is fiite, s the lst tem is specified. The othes e ifiite sequeces. Fo ech of the followig sequeces, stte whethe it is ithmetic, geometic, o eithe. ) 45, 5, 5, b) 5, 3,,, c), 8, 7, 64,, 000 d),,,,,, Aswe ) Geometic, becuse the commo tio is 3.
2 b) Aithmetic, becuse the commo diffeece d is. c) Neithe, becuse thee is t eithe commo diffeece o tio betwee tems. 3 (I fct, the ptte is tht.) d) Geometic, becuse the commo tio is. Agi, you c defie geometic sequece i oe of thee wys: by listig the tems, by givig ecusive defiitio, o by givig geel defiitio. Recusive Defiitios fo Geometic equeces Let s look t exmple. Give ecusive defiitio fo the sequece, 0, 50, 50, Aswe Recll tht ecusive defiitio hs two pts: listig the fist tem d givig the ptte. I this cse, the ptte is multiplyig the pevious tem by = 5 to get the ext tem. The ecusive defiitio is theefoe 5 Moe geelly, the ecusive defiitio fo y geometic sequece is Geel Fomule fo Geometic equeces <iset vlue hee> Let s exmie the pevious exmple i moe detil to see if we c ecogize y pttes d come up with geel fomul. Rewitig ech tem, we get, 0, 50, 50,... 3, 5, 5, 5,... o the 3 d tem equls the fist times 5 squed, the 4 th tem equls the fist times 5 cubed, d the th tem will equl the fist times 5 ised to the ( ) powe. Moe geelly, the th tem equls the fist tem times ised to the ( ) powe, mely
3 fo ll geometic sequeces. Wite geel fomul fo the sequece 3, 6,, Aswe This sequece is geometic with the fist tem 3 d commo tio. 3 The geel fomul is the tht = 3 . Wht is the 0 th tem i the sequece i the sequece 3, 6,,? Aswe This is the sme sequece fom the pevious exmple. We my the use the fomul we deived bove with = ,57,864 The 0 th tem is,57,864, which povides ice exmple fo how fst geometic sequeces c gow, eve fo smll vlues of. Wite geel fomul fo the sequece 8,, 8, 7,? Wht is the fifteeth tem i this sequece? The fiftieth? Aswe
4 o the geel fomul is d , espectively. d the fifteeth d fiftieth tems e Geometic eies Recll tht is the sum of the fist tems of seies. Let s look t how fomul fo is deived Let s tke tht lst expessio fo d multiply it by to get... The if we dd the ows fo d, we get sice ll of the tems i betwee these two ( d ) will ccel. The d The lst fomul bove is the fomul fo the sum of the fist tems fo y geometic seies.
5 Fid the sum of the fist 0 tems of the seies Aswe This is geometic seies with = 3 d =. We wt to fid The sum of the fist 0 tems is 3,45, ,45,75 Fid the sum of the fist foty tems of the seies Aswe This is geometic seies with = 8 d = We wt to fid The sum of the fist foty tems is um of Ifiite Geometic eies Let s tke look t the ifiite seies Wht hppes whe we ty to evlute this sum usig the fomul? We c put = ½, = ½, d = ito the fomul, but we will u ito odblock whe we ty to evlute (½).
6 Let s tke close look t the behviou of (½) fo lge vlues of. As gets lge, the fctio gets eve smlle. I fct, s ppoches, (½) will ppoch zeo. This is tue fo y povided tht <. (If you e ot fmili with the bsolute vlue bs, x, equivlet expessio is tht < <.) Recllig tht d lettig the tem go to zeo, the fo < < fo y ifiite geometic seies, povided tht meets the estictio bove. Let s ow evisit the seies tht stted this discussio,..., d evlute it i the followig exmple. Evlute Aswe This seies is geometic with = ½ d = ½. The The sum of this seies is. / / / / Evlute Aswe This seies is geometic with = 4 d = 3.
7 Evlute Aswe This seies is ideticl to the pevious oe except tht is ow egtive: = 4 d = Evlute Aswe This seies is geometic with = d = 3. You my ledy elize wht s goig o, but i cse you do t, let s ively put the vlues ito the fomul d see wht we get: 4 3 Wit! How c the sum of buch of positive umbe be egtive? The swe is tht ou estictio fo is tht it must be betwee d, but =.5. Becuse does ot stisfy the estictio, we cot use the bove fomul fo. Ideed, if you dd up buch of positive umbes tht e icesig s you go up, you c see tht the sum just keeps gettig bigge s we dd moe tems. You could the eithe sy tht the sum is ifiite (dicey) o does ot exist (sfe). But why is it sfe to sy does ot exist i the lst exmple? Let s look t thee sums: )
8 b) 8 7 c) Ech tem i ) is gettig moe positive, so the sum of tht sequece will be +. Ech tem i b) is gettig moe d moe egtive, so the sum of tht sequece will be. But i the lst tem, the sum oscilltes bck d foth: =, = 6, 3 =, 4 = 9.5, d so o. The sig of is eithe positive o egtive depedig o whethe the umbe of tems you ve dded is eve o odd. Rthe th debtig whethe ifiity is odd o eve (?!), we will just sy tht the sum does ot exist. Evlute j 7 j0 3. Aswe Ick! The best plce to stt is to figue out the fist few tems to detemie the ptte: whe j = 0, whe j =, whe j =, so ou sequece is 7, 9, 3, This is geometic with = 7 d = 3. The Evlute k. k 5
9 Aswe Oce gi, let s figue out the fist few tems to detemie the ptte: whe k = 5, k 5.5 whe k = 6, k 6 3 whe k = 7, k so ou sequece is.5, 3, 3.5. Wit! This is ithmetic! Not oly tht, but the umbes e icesig. o the sum will be ifiite, o if you pefe, the sum does ot exist. Repetig Decimls Let s exmie 0.7 i some detil to see wht we fid: But this is just the sum of ifiite seies with = 0.7 d = 0.. Rewitig d i fctio fom (you ll see why i miute) gives = 7 0 d = 0. The o 0.7 = 7/9. Iteestig! Fid exct fctio fo 0.6. Aswe
10 But this is just the sum of ifiite seies with = 6 0 d = 0. The o 0.6 = /3. Fid exct fctio fo 0.8. Aswe But this is just the sum of ifiite seies with = 8 00 d = 00. The o 0.8 = /. ummy Fo geometic sequece, the th tem is give by Fo geometic seies, the sum of the fist tems (th ptil sum) is Fo ifiite geometic seies, the sum is, povided tht < <.
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