Section 6.3: Geometric Sequences

Transcription

1 40 Chpter 6 Sectio 6.: Geometric Sequeces My jobs offer ul cost-of-livig icrese to keep slries cosistet with ifltio. Suppose, for exmple, recet college grdute fids positio s sles mger erig ul slry of $6,000. He is promised % cost of livig icrese ech yer. His ul slry i y give yer c be foud by multiplyig his slry from the previous yer by 0%. His slry will be $6,50 fter oe yer; $7, fter two yers; $7,59.4 fter three yers; d so o. Whe slry icreses by costt rte ech yer, the slry grows by costt fctor. I this sectio, we will review sequeces tht grow i this wy. Fidig Commo Rtios The yerly slry vlues described form geometric sequece becuse they chge by costt fctor ech yer. Ech term of geometric sequece icreses or decreses by costt fctor clled the commo rtio. The sequece below is exmple of geometric sequece becuse ech term icreses by costt fctor of 6. Multiplyig y term of the sequece by the commo rtio 6 geertes the subsequet term. Defiitio of Geometric Sequece A geometric sequece is oe i which y term divided by the previous term is costt. This costt is clled the commo rtio of the sequece. The commo rtio c be foud by dividig y term i the sequece by the previous term. If is the iitil term of geometric sequece d r is the commo rtio, the sequece will be, r, r, r,... Video Exmple : Overview of Geometric Sequeces. Give set of umbers, determie if they represet geometric sequece.. Divide ech term by the previous term.. Compre the quotiets. If they re the sme, commo rtio exists d the sequece is geometric.

2 Sectio 6. Geometric Sequeces 40 Exmple Is the sequece geometric? If so, fid the commo rtio..,, 4, 8, 6,... b. 48,, 4,,.... Divide ech term by the previous term to determie whether commo rtio exists The sequece is geometric becuse there is commo rtio. The commo rtio is. b The sequece is ot geometric becuse there is ot commo rtio. The grph of ech sequece is show below. It seems from the grphs tht both () d (b) pper hve the form of the grph of expoetil fuctio i this viewig widow. However, we kow tht () is geometric d so this iterprettio holds, but (b) is ot. If you re told tht sequece is geometric you ot hve to divide every term by the previous term to fid the commo rtio. You c choose y oe term i the sequece, d divide it by the previous term to fid the commo rtio.. Is the sequece geometric? If so, fid the commo rtio.. 5, 0, 5, 0,... 4 b. 00, 0, 4,, 5

3 404 Chpter 6 Writig Terms of Geometric Sequeces Now tht we c idetify geometric sequece, we will ler how to fid the terms of geometric sequece if we re give the first term d the commo rtio. The terms of geometric sequece c be foud by begiig with the first term d multiplyig by the commo rtio repetedly. For istce, if the first term of geometric sequece is, d the commo rtio is r = 4, we c fid subsequet terms by multiplyig 4 to get 8 the multiplyig the result 84 to get d so o. 4 ( )(4) 8 ( 8)(4) ( )(4) 8 The first four terms re {, 8,, 8}. Give the first term d the commo fctor, fid the first four terms of geometric sequece.. Multiply the iitil term, by the commo rtio to fid the ext term,. Repet the process usig to fid d the to fid 4 util ll four terms hve bee idetified.. Write the terms seprted by commos withi brckets. Exmple List the first four terms of the geometric sequece with 5 d r =. Multiply by to fid Repet the process, usig to fid d so o The first four terms re { 5, 0, 0, 40}.. List the first five terms of the geometric sequece with 8 d r =.

4 Sectio 6. Geometric Sequeces 405 Usig Recursive Formuls for Geometric Sequeces A recursive formul llows us to fid y term of geometric sequece by usig the previous term. Ech term is the product of the commo rtio d the previous term. For exmple, suppose the commo rtio is 9. The ech term is ie times the previous term. As with y recursive formul, the iitil term must be give. Recursive Formul for Geometric Sequece The recursive formul for geometric sequece with commo rtio r d first term is, r Give the first severl terms of geometric sequece, write its recursive formul.. Stte the iitil term.. Fid the commo rtio by dividig y term by the precedig term.. Substitute the commo rtio ito the recursive formul for geometric sequece. Exmple Write recursive formul for the followig geometric sequece. 6, 9,.5, 0.5,... The first term is give s 6. The commo rtio c be foud by dividig the secod term by the first term. r Substitute the commo rtio ito the recursive formul for geometric sequeces d defie..5 for r 6 The sequece dt poits follow expoetil ptter. The commo rtio is lso the bse of expoetil fuctio. You do ot hve to divide the secod term by the first term to fid the commo rtio. We c divide y term i the sequece by the previous term. It is, however, most commo to divide the secod term by the first term becuse it is ofte the esiest method of fidig the commo rtio.

5 406 Chpter 6. Write recursive formul for the followig geometric sequece. ì í, 4, 8 9, 6 î 7,... ü ý þ Usig Explicit Formuls for Geometric Sequeces Becuse geometric sequece is expoetil fuctio whose domi is the set of positive itegers, d the commo rtio is the bse of the fuctio, we c write explicit formuls tht llow us to fid prticulr terms. r Let s tke look t the sequece 8, 6, 7,44, 88,.... This is geometric sequece with commo rtio of d expoetil fuctio with bse of. A explicit formul for this sequece is 8 The grph of the sequece is show to the right. Explicit Formul for Geometric Sequece The th term of geometric sequece is give by the explicit formul: r Exmple 4 Give geometric sequece with d 4 4, fid. The sequece c be writte i terms of the iitil term d the commo rtio r., r, r, r, Fid the commo rtio usig the give fourth term. r Write the fourth term of the sequece i terms of 4 r Substitute 4 for 4 4 r Divide 8 r Solve for the commo rtio usig cube root r d r

6 Sectio 6. Geometric Sequeces 407 Fid the secod term by multiplyig the first term by the commo rtio. () 6 The commo rtio is multiplied by the first term oce to fid the secod term, twice to fid the third term, three times to fid the fourth term, d so o. The teth term could be foud by multiplyig the first term by the commo rtio ie times or by multiplyig by the commo rtio rised to the ith power. 4. Give geometric sequece with 4 d, fid 6. Exmple 5 (video exmple here) Write explicit formul for the th term of the followig geometric sequece., 0, 50, 50,... { } The first term is. The commo rtio c be foud by dividig the secod term by the first term. 0 5 The commo rtio is 5. Substitute the commo rtio d the first term of the sequece ito the formul. r 5 The grph of this sequece shows expoetil ptter. 5. Write explicit formul for the followig geometric sequece.,, 9, 7,... { }

7 408 Chpter 6 Solvig Applictio Problems with Geometric Sequeces I rel-world scerios ivolvig rithmetic sequeces, we my eed to use iitil term of 0 isted of. I these problems, we c lter the explicit formul slightly by usig the followig formul: r 0 Exmple 6 I 0, the umber of studets i smll school is 84. It is estimted tht the studet popultio will icrese by 4% ech yer.. Write formul for the studet popultio. b. Estimte the studet popultio i 00. The situtio c be modeled by geometric sequece with iitil term of 84. The studet popultio will be 04% of the prior yer, so the commo rtio is.04.. Let P be the studet popultio d be the umber of yers fter 0. Usig the explicit formul for geometric sequece we get P b. We c fid the umber of yers sice 0 by subtrctig = 7 We re lookig for the popultio fter 7 yers. We c substitute 7 for to estimte the popultio i 00. P The studet popultio will be bout 74 i A busiess strts ew website. Iitilly the umber of hits is 9 due to the curiosity fctor. The busiess estimtes the umber of hits will icrese by.6% per week.. Write formul for the umber of hits. b. Estimte the umber of hits i 5 weeks.

8 Sectio 6. Geometric Sequeces 409 Aswers. The sequece is ot geometric becuse b. The sequece is geometric. The commo rtio is 5.. 8,6,,, 9. for , P b. The umber of hits will be bout

9 40 Chpter 6 Sectio 6. Exercises. Wht is geometric sequece?. How is the commo rtio of geometric sequece foud?. Wht is the procedure for determiig whether sequece is geometric? 4. Wht is the differece betwee rithmetic sequece d geometric sequece? 5. Describe how expoetil fuctios d geometric sequeces re similr. How re they differet? For the followig exercises, fid the commo rtio for the geometric sequece. 6.,, 9, 7, 8, , 0.5, 0.5,,, 8.,, 8,, 8, For the followig exercises, determie whether the sequece is geometric. If so, fid the commo rtio. 9. 6,, 4, 48, 96, 0. 5, 5., 5.4, 5.6, 5.8,.,,,,,. 6, 8,, 5, 0, , 4, 0, 00, 500, For the followig exercises, write the first 5 terms of the geometric sequece, give the first term d commo rtio. 4. = 8, r = = 5, r = 5 For the followig exercises, write the first 5 terms of the geometric sequece, give y two terms = 64, 0 = = 5, 8 = 6.5 For the followig exercises, fid the specified term for the geometric sequece, give the first term d commo rtio. 8. The first term is, d the commo rtio is. Fid the 5 th term. 9. The first term is 6 d the commo rtio is. Fid the 4 th term. For the followig exercises, fid the specified term for the geometric sequece, give the first 4 terms. 0. {,, 4, 8, } Fid _.. {,, 9, 7, } Fid 7.

10 Sectio 6. Geometric Sequeces 4 For the followig exercises, write the first 5 terms of the geometric sequece.. = 486, =. = 7, = 0. For the followig exercises, write recursive formul for ech geometric sequece. 4. {, 5, 5, 5, } 5. {, 6, 8, 4, } 6. {4, 56, 4, 896, } 7. {0,, 0.9, 0.7, } 8. {0.6,.8, 5.49, 6.47, } 9. { 5, 0, 60, 60, } 0. {, 4, 8 9, 6 7, }. { 5, 8,, 8, } For the followig exercises, write the first 5 terms of the geometric sequece.. = 4(5 ). = ( ) For the followig exercises, write explicit formul for ech geometric sequece. 4. {, 4, 8, 6, } 5. {,, 9, 7, } 6. { 4,, 6, 08, } 7. {0.8, 4, 0, 00, } 8. {.5, 5, 0, 80, } 9. {, 4 5, 6 5, 64 5, } 40. {,, 8, 08, } 4. {,,, 9, } For the followig exercises, fid the specified term for the geometric sequece give. 4. Let = 4, =. Fid Let = ( ). Fid. For the followig exercises, fid the umber of terms i the give fiite geometric sequece. 44. {,, 9,, 87} 45. {,,,, 04 }

11 4 Chpter 6 For the followig exercises, determie whether the grph show represets geometric sequece For the followig exercises, use the iformtio provided to grph the first 5 terms of the geometric sequece. 48. =, r = 49. =, = 50. = 7(0.)