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1 Grde Level/Course: Algebr Lesso/Uit Pl Nme: Geometric Sequeces Rtiole/Lesso Abstrct: Wht mkes sequece geometric? This chrcteristic is ddressed i the defiitio of geometric sequece d will help derive the recursive formul. Studets will write the recursive d explicit formuls for geometric sequeces. Timefrme: clss periods Commo Core Stdrd F- BF.: Write rithmetic d geometric sequeces both recursively d with explicit formul use them to model situtios d trslte betwee the two forms. Notes: The Wrm- Up is o pge. A blck- lie mster of Exmple You Try is o provided o pge 0 for duplictio or use with projectio system. There re two forms of the recursive formul forms re used iterchgebly i this lesso. r d r. These two Istructiol Resources/Mterils: Wrm- Up Blck- lie mster Exmple visul id Idex Crds (optiol) Pge of 0//

2 Lesso: Thik- Pir- Shre: Describe the ptter i ech sequece. )... b)... TPS Aswers: ) Ech term is times the previous term. (Also the sequece is ot rithmetic.) b) Ech term is more th the previous term. (Also the sequece is rithmetic.) REVIEW from the Arithmetic Sequece Lesso: A sequece is list or ordered rrgemet of umbers figures or objects. The members which re lso elemets re clled the terms of the sequece. A geerl sequece c be writte s... where is the first term is the secod term d so o. The th term is deoted s. A geometric sequece is list of umbers i which the rtio of y term to the previous term is costt. The costt rtio is clled the commo rtio is deoted by r. r Exmple : Determie if the sequece is geometric. Justify your swer. Use the defiitio d check if ll rtios re the sme. Sequece... Coclusio Sice ll the rtios re costt (costtly ) the sequece is geometric d the commo rtio is. Pge of 0//

3 Try... Sice ll the rtios re differet d NOT the sme costt the the sequece is NOT geometric. Aother wy to represet the rtio symboliclly is to use rrows showig the multiplictio from oe term to the ext Thik- Pir- Shre: Expli to your prter wht the equtio C we rewrite this equtio i other form? r is used for d how to use it? Derive the Recursive Formul of Geometric Sequece Solve r for : r r r r r r (or r ) re equivlet equtios. The equtio r is clled the recursive formul of geometric sequece. It defies the ext term s the previous term times the commo rtio. It c be used to geerte the terms of geometric sequece oe term t time. Pge of 0//

4 Exmple : Give the geometric sequece ) Fid the ext term. Sice we kow the sequece is geometric there is commo rtio. Wht is it? Use r : r. Usig 00 r r 0. 0 Method Sice we kow the fourth term d the commo rtio we c use the recursive formul to fid the fifth term. Use r. Substitute d simplify Recursive Formul: Use the equtio derived o the previous pge. The recursive formul is used to fid the ext term i the sequece by multiplyig the previous term by the commo rtio. r 0 The recursive formul is 0 *Remember there re two prts to this formul. Method Rewrite ech term i terms of the first term d the commo rtio ? Explicit Formul: Use the ptter to fid y term. The explicit formul is used to fid y term i the sequece without kowig the previous term *Use more terms if eeded to get studets to see this ptter. Thik- Pir- Shre: Which formul(s) c be used to fid 0? Which formul would be most efficiet? Justify your swer. Both formuls c be used to fid the 0 th term. Method is ot very time efficiet s you would eed to fid ll the terms ledig up to 0. Method is the most direct pproch sice you oly eed to kow the vlue of which i this cse is 0. b) Fid 0. If 0 the Therefore. Pge of 0//

5 Discuss: Should we evlute or leve it s is? Refer bck to the defiitio of geometric sequece d geerlize : A geometric sequece c be writte s r r r r r... r r... where is the first term or iitil coditio d r is the commo rtio. Like rithmetic sequeces geometric sequeces lso hve recursive d explicit formuls. The formuls for rithmetic sequeces re provided for review d pplictio. Type of Sequece Recursive Formul (or rule) Explicit Formul (or rule) Arithmetic + d + ( )d The commo differece is d. Geometric where is give r r The commo rtio is r. where is give Exmple : Write the recursive d explicit formuls for the sequece... Thik- Pir: Wht kid of sequece is...? Which formuls do we use? The sequece is geometric so use the geometric formuls. Commo Rtio Recursive Formul (or rule) Explicit Formul (or rule) r Use r d replce the r vlue d stte the first term. Use r d replce the r vlue d the first term. r or where Pge of 0//

6 Exmple b (optiol): Use both formuls to fid. Use Recursive Formul (or rule) where d. Use Explicit Formul (or rule) where. Idetify the terms up to :?? st : Fid d : Fid 0 ( ) ( ) TRY (with Solutios): Decide whether the sequece is rithmetic geometric or either. Fid the ext term. The write the recursive d explicit formuls. Sequece Next term Commo Rtio or Differece Recursive Formul (or rule) Explicit Formul (or rule) Prter A Prter B Techer s Choice (Geometric) (Geometric) (Arithmetic) r ( ) where r ( ) d + where ( ) where ( ) ( )( ) + or 7 Pge of 0//

7 Exmple : Suppose you drop bll from height of 00 cm. It bouces bck to 80% of its previous height. ) About how high will the bll go fter its fifth bouce? Iitil height of bll: After first bouce: 80% of 00 cm After d bouce: 80% of 80 cm 00 cm 0.80(00 cm) 80 cm 0.80(80 cm) cm After rd bouce: 80% of cm 0.80( cm). cm After th bouce: 80% of. cm 0.80(. cm) 0.9 cm After th bouce: 80% of 0.9 cm 0.80(0.9 cm).78 cm Techer Note: Omit the uderlied vlues d prompt studets to idetify the correct vlue tht belogs i ech blk. Therefore the bll will reboud bout.8 cm fter the fifth bouce. Thik- Pir: Wht kid of sequece is creted by these heights? How do you kow? Expli wht the 00 cm d the 80% represet. b) Write the recursive d explicit formuls for the geometric sequece geerted by these heights. To write both formuls idetify the commo rtio d the first term: r 0. 8 d 00 Recursive Formul: ( ) 0.8 where 00 Explicit Formul: ( ) TRY: Model the situtio below usig recursive d explicit formul. At the begiig of experimet there re 00 bcteri coloies. The umber of coloies doubles ech hour. *Note: This problem is from the wrm- up. Recursive Formul: where 00 Explicit Formul: ( ) 00 Pge 7 of 0//

8 Thik- Pir- Shre/Discussio Questios: Why is it ecessry to idetify the first term i the recursive formuls? If the first term is ot idetified the the formul represets y sequece tht hs the sme commo rtio. For exmple ( ) represets the sequeces d How do geometric sequeces with positive commo rtio compre to geometric sequeces with egtive commo rtio? The terms of geometric sequece with positive commo rtio re the sme sig (ll positive or ll egtive) wheres the terms of geometric sequece with egtive commo rtio lterte sigs. How re geometric sequeces similr to expoetil fuctios? The explicit formuls of geometric sequeces re expoetil fuctios. How is geometric sequece differet from expoetil fuctio? A expoetil fuctio is cotiuous d the domi is ll rel umbers. A geometric sequece is collectio of poits tht re ot coected which mke it ot cotiuous d the domi is ll turl.... umbers { } Is geometric sequece fuctio? Yes geometric sequece is fuctio whose domi is ll turl umbers {... }. Therefore the explicit formul of geometric sequece c be writte i fuctio ottio. Arithmetic d Geometric Sequeces d Fuctio Nottio: Sequece Recursive Formul i Fuctio Nottio Explicit Formul i Fuctio Nottio Arithmetic f ( ) f ( ) + d with f ( ) give f ( ) f ( ) + ( )d Geometric f ( ) f ( ) r with f ( ) give ( ) ( ) f f r Pge 8 of 0//

9 Exit Ticket Optios: Extesio Activity/Optio #:. Ech studet cretes their ow geometric sequece d writes it o oe side of idex crd.. Ech studet writes the recursive d explicit formuls for their sequece o the other side of crd.. Studets write their mes o oe side of the idex crd d submit them to the techer.. Techer checks crds for ccurcy.. Techer returs crds to studets o differet dy so they c prticipte i the Quiz- Quiz- Trde Activity. Techer tells studets to std up d pir up. Prter A quizzes B. Prter B swers. Prter A prises or coches. Prters switch roles. Prters trde crds. After techer specified time repet the steps bove with ew prter. Optio #: Determie if ech sttemet is true or flse. ) The sequece is geometric. True Flse ) The sequece 8... is geometric. True Flse ) The recursive formul 80 represets the sequece True Flse ) The explicit formul ( ) represets the sequece... True Flse ) The recursive formul ( ) ( ) represet differet sequeces. d the explicit formul True Flse ANSWERS: F T T F F Pge 9 of 0//

10 Type of Sequece Recursive Formul (or rule) Explicit Formul (or rule) Arithmetic + d + ( )d Hve commo differece d. Geometric where is give r r Hve commo rtio r. where is give Sequece Next term Commo Rtio or Differece Recursive Formul (or rule) Explicit Formul (or rule) Circle Sequece Type: Geometric Arithmetic Neither... Circle Sequece Type: Geometric Arithmetic Neither... Circle Sequece Type: Geometric Arithmetic Neither Circle Sequece Type: Geometric Arithmetic Neither Pge 0 of 0//

11 Exmple Visuls Pge of 0//

12 Wrm- Up CCSS: F- LE. y CCSS: F- BF. Use the grph below to determie if ech sttemet is true or flse. Give the rithmetic sequece 8...: A) f ( x) x + True Flse B) f ( x) + x True Flse ) Describe why the sequece is rithmetic. x C) f ( x) True Flse b) Idetify the commo differece. D) f ( x) x+ True Flse c) Write the recursive formul. E) f ( x) 0 True Flse d) Write the explicit formul. Fid the vlue of f ( ). CCSS: F- LE. cotiued Curret: x y At the begiig of experimet there re 00 bcteri coloies. The umber of coloies doubles ech hour d is recorded. Complete the tble. Redig Number of Coloies x 00 0 Pge of 0//

13 Wrm- Up Solutios CCSS: F- LE. y CCSS: F- BF. Use the grph below to determie if ech sttemet is true or flse. Give the rithmetic sequece 8...: A) f ( x) x + True Flse B) f ( x) + x x C) f ( x) D) f ( x) x+ E) f ( x) 0 f ( ) True True True True Flse Flse Flse Flse ) The sequece is rithmetic becuse there is commo differece betwee cosecutive terms. b) The commo differece is. c) where d) ( ) or + CCSS: F- LE. cotiued Curret: x y At the begiig of experimet there re 00 bcteri coloies. The umber of coloies doubles ech hour d is recorded. Complete the tble. Redig Number of Coloies x Pge of 0//

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