PREFACE. Synergy for Success in Mathematics 10 is designed for Grade 10 students. The textbook

Transcription

1 Syergy for Success i Mthemtics 0 is desiged for Grde 0 studets. The textbook cotis ll the required lerig competecies d is supplemeted with some dditiol topics for erichmet. Lessos re preseted usig effective Sigpore Mth strtegies tht re iteded for esy uderstdig d grsp of ides for its trget reders. Vrious exercises re lso provided to help the lerers cquire the ecessry skills eeded. The book is orgized with the followig recurrig fetures i every chpter: Lerig Gols Itroductio Historicl Note This gives the specific objectives tht re iteded to be chieved i the ed. The reder is give bird's eye view of the cotets. A brief historicl ccout of relted topic is icluded givig the reder wreess of some importt cotributios of some gret mthemticis or eve stories of gret chievemets relted to mthemtics. Method/Exm Notes These dditiol tools help studets recll importt iformtio, formuls, d shortcuts, eeded i workig out solutios. Exmples Ehcig Skills Likig Together Chpter Test Chpter Project Mkig Coectio PREFACE Step-by-step d detiled demostrtios of how specific cocept or techique is pplied i solvig problems. These re prctice exercises foud fter every lesso, tht will cosolidte d reiforce wht the studets hve lered. This visul tool c help the studets relize the coectio of ll the ides preseted i the chpter. This is summtive test give t the ed i preprtio for the expected ctul clssroom exmitio cotiig the topics icluded i the chpter. A chllegig tsk is desiged for the lerer givig him opportuity to use wht he/she hs lered i the chpter. This my be mipultive type of ctivity tht is specificlly chose to ehce uderstdig of the cocepts lered i the chpter. The studets re exposed to fcts d iformtio tht coect mthemtics d culture. This is for the purpose of lettig the lerers pprecite the subject becuse of tgible or true-tolife stories tht show how mthemtics is useful d relevt. Every effort hs bee mde i order for ll the discussios i this book to be cler, simple, d strightforwrd. This book lso gives opportuities for the reders to see the beuty of mthemtics s essetil tool i uderstdig the world we live i. With this i mid, pprecitio of mthemtics goes beyod seeig; relizig its criticl pplictio to decisio mkig i life completes the purpose of kowig d uderstdig mthemtics.

2 Tble of C tets CHAPTER SEQUENCES AND THE BINOMIAL THEOREM Itroductio... Historicl Note.... Ptter d Sequeces...3 The Geerl Term of Sequece... 3 Fiite d Ifiite Sequece Arithmetic Sequece...9 Defiitio of Arithmetic Sequece... 9 The Geerl Term of Arithmetic Sequece... Arithmetic Series... 5 Grphig Arithmetic Sequece... 8 The Arithmetic Me of Arithmetic Sequece... Applictios of Arithmetic Sequece Geometric Sequece...9 The Geerl Term of Geometric Sequece... 3 Geometric Series Ifiite Geometric Series Grphig Geometric Sequece Geometric Me Applictio of Geometric Sequece Hrmoic d Fibocci Sequece...58 Hrmoic Sequece The Geerl term of Hrmoic Sequece Hrmoic Me... 6 Applictio of Hrmoic Sequece The Fibocci Sequece The Geerl Term of Fibocci Sequece The Golde Rtio The Biomil Theorem Biomil Theorem The rth term i Biomil Expsio... 80

3 Likig Together...86 Chpter Test Chpter Project Mkig Coectio CHAPTER POLYNOMIALS Itroductio Historicl Note Polyomils Review o Polyomils Review o Opertios Ivolvig Polyomils Fctorig d Solvig Polyomil Equtios... 8 Evlutig Polyomil Fuctio 0 Compositio of Fuctios.3 Polyomil Theorems Zeros of Polyomil Fuctios... 4 Descrtes' Rule of Sigs 4.5 Grphs of Polyomil Fuctios Costt Fuctio Lier Fuctio Qudrtic Fuctio Cubic Fuctio Specified Domi Likig Together...73 Chpter Test...74 Chpter Project...79 Mkig Coectio...80 CHAPTER 3 PLANE COORDINATE GEOMETRY Itroductio 8 Historicl Note 8 3. The Distce Betwee Two Poits i the Crtesi Coordite Ple 83 Review o the Crtesi Coordite System 83 Icremet i x d y 86 Distce Betwee Two Poits i the Crtesi Ple Review o Rdi Mesure d The Uit Circle 06 Review o the Midpoit Formul 06 Divisio of Lie Segmet 0

4 3.3 Review o the Slope d Equtio of Lie 3 Review o the Slope of Lie 3 Review o the Equtio of Lie 7 Likig Together 44 Chpter Test 45 Chpter Project 49 Mkig Coectio 50 CHAPTER 4 CIRCLES Itroductio 87 Historicl Note Tgets 89 Circle 89 Tget Cetrl Agles, Arc, d Chords 309 Cetrl Agles d Arcs 309 Arcs d Chords Iscribed Agles d Itercepted Arcs Other Agle Properties of Circle 344 Agle Formed by Chord d Tget 344 Lies Itersectig Iside or Outside Circle Legths of Segmets i Circle Equtio of Circles 38 Likig Together 40 Chpter Test 40 Chpter Project 405 Mkig Coectio 406 CHAPTER 5 MEASURES OF POSITION Itroductio 407 Historicl Note Smples d Dt 409 Revisio o Sttistics 409 Types of Smplig 4 Revisio o Types of Dt 47 Levels of Mesuremet 49

5 Methods of Dt Collectio Sttisticl Represettios 48 Revisio o Sttisticl Grphs d Chrts 48 Dot Digrm 33 Stem-d-Lef Disply Mesures of Positio d Box-d-Whisker Plot 447 Percetile 447 Qurtile 450 Box-d-Whisker Plot 454 Decile 464 Likig Together 47 Chpter Test 473 Chpter Project 476 Mkig Coectio 478 CHAPTER 6 PROBABILITY Itroductio 479 Historicl Note Review o Fudmetl Coutig Priciple Permuttio 495 Fctoril Nottio 495 Permuttio Combitio Probbility of Evets 5 The Uio d Itersectio of Evets 57 Likig Together 57 Chpter Test 58 Chpter Project 53 Mkig Coectio 53 Glossry 533 Idex 544 Bibliogrphy 550

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7 SEQUENCES AND THE BINOMIAL THEOREM Lerig Gols At the ed of the chpter, you should be ble to:. Fid d geerte ptters.3 Describe d illustrte geometric sequece, give exmples of geometric sequeces, differetite fiite geometric sequece from ifiite geometric sequece, differetite rithmetic sequece from geometric sequece, fid the terms of geometric sequece icludig the geerl th term of the sequece, d fid the sum of terms of give geometric sequece. The piepple is tropicl plt. It is cocetrted i tropicl regios of the world. Sevety-five percet of piepple supplies ll over the world re produced i Thild, the Philippies, Brzil, Idi, Cost Ric, Nigeri, Mexico, d Idoesi. The fruit is med piepple becuse of its similrity to pie coe. Whe you look t the scles of the piepple, you see sets of spirl hexgos. There re three distict sets of hexgos. A set of five prllel spirls sceds t shllow gle to the right. The, set of eight prllel spirls rises more steeply th the previous set does d sceds to the left. Filly, set of 3 prllel spirls sceds very steep to the right. Notice tht the three sets represet segmet of the Fibocci sequece. Fibocci umber ptters, which re discussed i this chpter, coti terms just like the ptters o the scle of the piepple..4.5 Describe d illustrte rithmetic sequece, give exmples of rithmetic sequeces, fid the terms of rithmetic sequece icludig the geerl th term of the sequece, d fid the sum of terms of give rithmetic sequece Illustrte other types of sequeces (e.g., hrmoic d Fibocci) Solve problems ivolvig sequeces d series, icludig the biomil theorem

8 Historicl Note Leordo Piso Bigollo (70 50), lso kow s Leordo Fibocci or most commoly Fibocci, ws Itli mthemtici. He ws cosidered by some s the most tleted wester mthemtici of the Middle Ages. Fibocci is best kow for itroducig the Hidu-Arbic umerl system i Europe through his Liber Abci, which mes Book of Clcultio i 0. I his book, Fibocci itroduced the Modus Idorum (method of the Idis), which is trditiolly kow s Arbic umerls. He believed tht rithmetic with Hidu-Arbic umerls ws esier d more efficiet th with Rom umerls. He is lso kow for his umber sequece clled the Fibocci umbers.

9 Syergy for Success i Mthemtics Chpter. Ptters d Sequeces A umber sequece is set of umbers rrged i such wy tht ech successive umber follows the precedig umber ccordig to ptter or rule. A rule defies which wy to produce the vlue of ech term of sequece. Without defiite rule, sequece cot be formed. Defiitio of Sequece A sequece is set of umbers or object rrged i specific order. Ech umber or object i the sequece is clled term. The Geerl Term of Sequece Cosider the list of umbers below., 5, 9, 3, 7,, 5 The first umber is ; the secod is 5; the third is 9, d so o. The list of umbers is exmple of sequece sice it follows certi rule. The rule of the give sequece is to dd four to every term to fid the successive umber. Addig 4 to the first term results i 5, which is the secod term. Cotiue the process util the sum obtied is 5, which is the lst term. Exm Note The term of sequece c lso be clled the elemet or member of sequece Every elemet i the sequece is clled the term of the sequece. The ottio represets the th term or geerl term of sequece. The ottio represets the first term of the sequece; represets the secod term; 3 represets the third term; d so o. Thus, i the give sequece bove, =, = 5, = 9, d so o. 3 Exm Note The expressio represets the terms t the th positio i the sequece. The subscript is the term umber. 3

10 Fiite d Ifiite Sequeces The umber of terms i sequece is clled the legth of the sequece. To determie the legth of give sequece, fid out if it is fiite or ifiite sequece. A fiite sequece is sequece tht hs limited umber of terms. The terms re deoted by,, 3, where is turl umber. A ifiite sequece is sequece tht hs edless umber of terms. To idicte ifiite sequece, you my use ellipsis. The followig sets of umbers re exmples of fiite d ifiite sequeces. () 3, 7,, 5, 9 The set bove is fiite sequece. There re five terms: the first term is 3; the secod term is 7; d so o. The fifth d lst term is 9. (b),, 3, 4, 5, 6, ¼ The set bove is ifiite sequece. There re edless umber of terms: the first term is ; the secod term is ; the third is 3; d so o. The ellipsis fter 6 idicte tht there re more terms fter the term 6. You c fid the terms i sequece by substitutig positive itegers, strtig from, for the vrible i the geerl term of the sequece. Exm Note The sig fter give term is clled ellipsis. It idictes tht the sequece hs ifiite umber terms. Exmple Give the first five terms of the sequece whose th term is give by ech the followig formuls. () = + 5 (b) = (c) = ( ) () 4

11 Syergy for Success i Mthemtics Chpter SOLUTION () The first five terms of the sequece whose geerl term is = + 5 c be foud by substitutig,, 3, 4, d 5 for the vrible. First term: Secod term: Third term: Fourth term: Fifth term: = ()+ = 5 7 = ()+ = 5 9 = ()+ = 3 5 = 4 ( )+ 5= 3 = ()+ = Thus, the first five terms of the sequece re 7, 9,, 3, d 5. Method Note Substitute the turl umber for the vrible i the formul to fid the th term of the sequece. (b) = ( ) () First term: Secod term: Third term: Fourth term: Fifth term: ( ) ()= = = ( ) ()= 3 = ( ) ()= = ( ) ( 4)= 4 5 = ( ) ()= 5 5 (c) Thus, the first five terms of the sequece re -,, -3, 4, d -5. = First term: = = Secod term: = Third term: 3 = 3 Fourth term: 4 = 4 Fifth term: 5 = 5 Thus, the first five terms of the sequece re,, 3, 4, d 5. 5

12 You c lso fid the terms of sequece by ivestigtig its ptter d rule. Exmple Determie the ptter of ech give sequece. The, fid the ext term i the sequece. () 5, 0, 5, 30, (b) 4, 8, 6, 3, (c) -5, 5, -5, 5, (d) -8, -5, -,, SOLUTION () (b) (c) The ptter is ddig 5 to the previous term to get the ext term. Therefore, the ext term is 35. The ptter is multiplyig the term by to fid the ext term. Therefore, the ext term is 64. The costt multiplier is -. Thus, the ext term is -5. (d) The costt icrese is 3. Thus, the ext term is 4. Exmple 3 Fid the explicit formul for ech sequece. () 7, 0, 3, -4, (b) -40, -3, -4, -6, (c) 9, 9, 9, 39, (d) -4,, -36, 08, SOLUTION () = 4 7 (b) = (c) = (d) ( )( ) = 4 3 6

13 Syergy for Success i Mthemtics Chpter ENHANCING SKILLS A Give the first five terms of the sequece whose th term or geerl term is give by ech of the followig formuls. () = 5 + (6) ( ) + = () = 3 (7) = (3) = + (8) = 3 ( ) + ( ) (4) = 4 (9) = ( ) (5) = 3 (0) = + ( ) () 7

14 B Determie the ptter of ech give sequece d the fid the ext two terms of the sequece. (), 6, 0, (6) -5, -, -9, Ptter: Ptter: Next two terms: Next two terms: () 8, 3, 8, Ptter: (7), 8, 3, Ptter: Next two terms: Next two terms: (3) 3, 9, 7, Ptter: (8) 6, 8, 54, Ptter: Next two terms: Next two terms: (4) 4, 6, 64, Ptter: (9) -, 4, -8, Ptter: Next two terms: Next two terms: (5) -4, -8, -, Ptter: (0) 3, -9, 7, Ptter: Next two terms: Next two terms: 8

15 Syergy for Success i Mthemtics Chpter. Arithmetic Sequece The dt below show mothly ret of prtmet i Quezo City from yer 00 to 04. Yer Mothly retl fee i ( ) 5,500 6,000 6,500 7,000 7,500 Strtig t 5,500 i 00, the mothly ret icreses by 500 yerly. By yer 04, the ret becomes 7,500. The mothly ret the from 00 to 04, with differece of 500 betwee two cosecutive yers, forms sequece clled the rithmetic sequece. Defiitio of Arithmetic Sequece The vlues 5,000, 6,000, 6,500. 7,000, d 7,500 form rithmetic sequece. The vlue 500, the costt differece betwee the cosecutive terms of the sequece, is clled the commo differece. Defiitio of Arithmetic Sequece A rithmetic sequece is umber sequece whose every two cosecutive terms hve costt differece. The differece betwee the cosecutive terms is clled the commo differece. The first term i rithmetic sequece is deoted by while the commo differece is deoted by d. The commo differece is idetified by subtrctig two cosecutive terms i the sequece. Below re exmples of rithmetic sequeces. Give re the first term d commo differece of ech sequece. Method Note You c express the terms of rithmetic sequece give the vlues of d d s follows.,( + d),( + d),( + 3d), () 5, 7, 9,, = 5 d = () 00, 90, 80, 70, 60, = 00 d = -0 (3) 50, 00, 50, 00, = 50 d = 50 9

16 (4) 0, 9, 8, 7, 6, = 0 d = - (5) 3,,,, = d = Exmple Tell whether ech give sequece is rithmetic or ot. If the sequece is rithmetic, idetify its first term d the commo differece. (), 7,, 7, (b), 5, 7, 8, (c) 500, 400, 300, 00, (d) 3 4 5,,,, 3 3 (e),,,, 4 SOLUTION (), 7,, 7, = d = 5 (b), 5, 7, 8, The sequece is ot rithmetic becuse there is o commo differece. (c) 500, 400, 300, 00, = 500 d = 00 (d) 3 4 5,,,, 3 3 = 3 d = 3 (e),,,, 4 The sequece is ot rithmetic becuse there is o commo differece. 0

17 Syergy for Success i Mthemtics Chpter The Geerl Term of Arithmetic Sequece Cosider rithmetic sequece whose first term is d whose commo differece is d. The first five terms of the sequece re computed s follows. First term ( ): Secod term ( ): + d Third term ( 3 ): ( + d)+ d= + d Fourth term ( 4 ): ( +d)+ d= + 3d Fifth term ( 5 ): ( +3d)+ d= + 4d Followig the ptter bove, you c costruct equtio to fid the th term (or geerl term) of the give sequece.. The Geerl Term of Arithmetic Sequece The geerl term of rithmetic sequece is deoted by = + ( - ) d where is the geerl term of sequece, is the first term, is the positio of the term, d d is the commo differece. If the first term d the commo differece of rithmetic sequece re give, the geerl term of the sequece c be foud. To fid the vlue of the geerl term, we my substitute the give iformtio i the formul. The commo differece of rithmetic sequece c be foud usig y two cosecutive terms of the sequece. d= = 3 = 4 3 =

18 Exmple Fid the 0th term of rithmetic sequece whose first term is 8 d whose commo differece is 7. SOLUTION Use the formul for the geerl term of rithmetic sequece to fid the 0th term. = + d 0 0 ( ) = 8+ ( 0 )() 7 = 8+( 9)() 7 = = 7 Thus, the 0th term of the rithmetic sequece is 7. Without usig the formul for the geerl term of rithmetic sequece, c you still fid the vlue of certi term? Yes, you c by simply listig ll the terms of the sequece. The result i Exmple c be verified by listig the first 0 terms of the rithmetic sequece. Use the first term d the commo differece to fid the ext terms. Add 7 to the first term 8 to fid the secod term. Add 7 to the secod term to fid the third term. Do this util you rech the vlue of the 0th term. Therefore, the first 0 terms of the sequece re 8, 5,, 9, 36, 43, 50, 57, 64, d 7. Exmple 3 Fid the 7th term of rithmetic sequece whose first term is 85 d whose commo differece is -9. SOLUTION Apply the formul for fidig the geerl term of rithmetic sequece to fid the 7th term. = + ( ) d 7 7 = 85+ ( 7 ) ( 9) = 85+( 6) ( 9) = = 3 Thus, the 7th term of the rithmetic sequece is 3. Exm Note You c check the missig term by listig the umbers from the first term up to the geerl term usig the commo differece.

19 Wht if the rithmetic sequece is give, d you re sked to fid the defiig rule of the sequece? You c costruct equtio or formul to fid the geerl term wheever the rithmetic sequece is give. Exmple 4 Give the formul for the geerl term of the rithmetic sequece 5, 9, 3, 7, SOLUTION { } Syergy for Success i Mthemtics Chpter The first term i the give rithmetic sequece is 5. Thus, = 5. To get the commo differece, pply d=. d= = = = 9 5 = 4 Substitute the vlues 4 d 5 for d d, respectively, i the formul for the geerl term of rithmetic sequece. Costruct the resultig equtio or formul tht defies the geerl term of the sequece. = + ( ) d = 5+ ( ) = +( )( ) Method Note To fid the vlue of the commo differece d of rithmetic sequece, subtrct y two cosecutive terms; tht is, d = - -. Exmple 5 Fid the formul tht defies the geerl term of rithmetic sequece whose secod term is 9 d whose seveth term is 39. SOLUTION To determie the geerl term of rithmetic sequece, fid the first term d the commo differece d. The secod term of the rithmetic sequece is 9. Thus, = 9. Apply the formul to fid the geerl term of rithmetic sequece. 3

20 = + d = + d 9= + d ( )( ) ( )( ) + d= 9 () The seveth term of the sequece is 39. Apply the formul for the geerl term. = + d = + 7 d 7 39= + 6d ( )( ) ( )( ) + 6d= 39 () You ow hve two lier equtios with two ukows. Use equtios () d () to fid the vlues of d d. () + d= 9 () + 6d= 39 To elimite, multiply the secod equtio by -. ( ) ( + 6d)=( 39) ( ) 6d = 39 (3) Now, dd equtios () d (3) to elimite. + d= 9 6d= 39 The commo differece is 6. 5d = 30 5d 30 5 = 5 5d = 5 d = 6 ( )( ) Substitute the vlue of d i either of the equtios () d (), d fid the vlue of the first term. 4

21 + d= 9 + 6= 9 = 3 Use equtio (). Substitute d for 6. Subtrct 6 from both sides. Thus, the first term of the rithmetic sequece is 3. Now usig the vlue of d d, costruct formul tht defies the give sequece. Hece, the geerl term of the give rithmetic sequece is = + d = 3+ 6 = = 6 3. ( ) ( )( ) Syergy for Success i Mthemtics Chpter Arithmetic Series Cosider the followig rithmetic sequece. 35, 53, 7, 89, 07, 5, 43, 6, 79, 97 Fid the sum of the first te terms of the sequece. I solvig the problem, you my list ll the ddeds d dd directly = 60 If you were to fid the sum of the first 00 terms of rithmetic sequece, would you use the sme method tht is, listig the terms d ddig them directly? The sum of the first terms is deoted by S ; is the first term; d d is the commo differece. You c represet the sum S usig equtios () d (). S = + ( + d)+ + + ( ) d + + d () S = + ( ) () Add the two equtios () d (). 5

22 ( ) ( ) S = + + d d d S = + d d d ( ( ) ) + + ( ) ( ) ( ) + ( + ( ) ) S = ( + ( ) d)+ ( + ( ) d)+ + ( + ( ) d)+ + ( ) d ( ) + ( ) Simplify the sum. Divide both sides by. =( )( + ) S S = () + ( ) Method Note To sum up the terms of rithmetic sequece, we c use the followig formul. + ( + d)+ ( + d)+ ( + 3d)+ Arithmetic Series The sum S of the first terms of rithmetic sequece is S = + ( ) where is the first term d is the lst term. Exm Note A rithmetic series is the sum of the terms of rithmetic sequece. You c tke the sum of fiite umber of terms of rithmetic sequece. Exmple 6 Fid the sum of the first 50 terms of the followig rithmetic sequece., 5, 8,, 4, SOLUTION To fid the sum of the first 50 terms, determie first the commo differece d. d= = 5 = 3 Thus, the commo differece d is 3. Next, fid the 50th term of the give rithmetic sequece. 6

23 Syergy for Success i Mthemtics Chpter = + d 50 ( ) = + ( 50 )() 3 = +( 49)() 3 = + 47 = 49 Thus, the 50th term of the sequece is 49. Sice the 50th term is idetified, use it to fid the sum of the first 50 terms. S= ( + ) S = 50 ( + ) = 50 ( + ) 49 = ( 5)( 5) = 3775 Therefore, the sum of the first 50 terms of the rithmetic sequece is I the previous exmple, the th term is foud before gettig the sum S. You c lso fid the sum of the terms without determiig the vlue of. Sice = + ( - ) d, you c substitute ( + ( - )d) for i the formul for S. The formul for S c be expressed s follows. S= ( + ) S = + + ( ) d Substitute + ( - ) for. S = + ( ) d Simplify the equtio. Thus, the sum of the first terms of rithmetic sequece c be lso foud by usig the formul S = + ( ) d. 7

24 Exmple 7 Determie the sum of the first 00 odd umbers from zero. SOLUTION The give rithmetic sequece is the first 00 odd umbers. Hece, the sequece is, 3, 5, 7, The commo differece of the sequece is. Substitute 00 for, for d, d for = 00, d =, d = i the followig formul. S = + ( ) d S = 00 ()+( ) = ( 50) +( 99 ) =( )( + ) =( 50)( 00) = Thus, the sum of the first 00 odd umbers is Grphig Arithmetic Sequece Remember tht for you to locte d grph poit, you eed to hve coordites which specify the positio or loctio of poit. The x-coordite is the first umber i the coordites d represets the distce of the poit from the y-xis. The y-coordite is the secod umber i the coordites d represets the distce of the poit from the x-xis. The coordites of poit o Crtesi ple re represeted by ordered pir. (, b) x-coordite y-coordite The grph of sequece is formed by collectig ll poits with coordites (, ) where =,, 3,... d correspods to the th term of the sequece. 8

25 For exmple, grph the ifiite rithmetic sequece whose terms re { 3, 7,, 5, } Idetify ll ordered pirs (, ) tht stisfy the rithmetic sequece d list them i the tble. b (, b) 3 (, 3) 7 (, 7) 3 ( 3, ) 4 5 ( 4, 5) (, ) Syergy for Success i Mthemtics Chpter The grph of the give sequece is show below. 4 y (4, 5) 0 (3, ) 8 6 (, 7) 4 (, 3) x 9

26 Exmple 8 Grph the followig rithmetic sequeces. { } () 6, 4,, 0,, (b) { 05., 75., 5, 75., 95., } SOLUTION () The first five poits of the grph of the rithmetic sequece re (, 6), (, 4), ( 3, ), ( 4, 0), d ( 5, ). The grph of the sequece is show below with two more poits, ( 6, 4) d ( 7, 6), which stisfy the sequece. Method Note 6 4 y (5, ) (6, 4) (7, 6) Use the formul for the geerl term of rithmetic sequece to determie the poits tht re icluded i the grph of the give sequece. (4, 0) x (3, -) (, -4) (, -6) (b) The first five poits of the grph of the rithmetic sequece re (, 05. ), (, 75. ), ( 3, 5), ( 4, 75. ), d ( 5, 95. ). Cotiuig the ptter, the grph of the sequece is show i the followig figure. y 4 (7, 4) Method Note Collect smple poits for the grph of the first give sequece (6,.75) (5, 9.5) (4, 7.5) (3, 5) (,.75) 0 (, 0.5) x

27 The Arithmetic Me of Arithmetic Sequece Syergy for Success i Mthemtics Chpter Cosider two umbers d b. Oe or more umbers c be iserted betwee d b to form rithmetic sequece with my terms. The iserted umbers re clled the rithmetic mes of the give umbers. To fid the rithmetic me betwee two umbers d b such tht the three umbers form rithmetic sequece, you my use the formul for fidig the verge of two umbers. Exm Note I sttistics, the term rithmetic me lso refers to the sum of the terms of set of umbers divided by the umber of terms. It is lso clled me or verge. Let the rithmetic sequece be {, x, b}. The two terms of the sequece re d b, d x is the rithmetic me. Let d be the commo differece of the sequece. Hece, the commo differece c be expressed s d= x or d= b x. Now, fid the vlue of x i the followig equtio. d= d x = b x x= + b b x = + Apply substitutio. Combie similr terms. Divide both sides by. Therefore, the rithmetic me of two umbers d b is give by + b. The three terms of the sequece re, + b., d b. Fid the rithmetic me of the followig sequece.,, 6 The vlues of d b re to d 6, respectively. Let x be the rithmetic me of the sequece. To fid the vlue of the rithmetic me, clculte the verge d 6, of the first d the lst terms of the sequece. Hece, b x = + = = = 4.

28 Exmple 9 Isert four rithmetic mes betwee -0 d 5. SOLUTION Plce blks betwee -0 d 5. The blks represet the four rithmetic mes. -0,,,,, 5 Clerly, there re six terms i the sequece. Let,, 3, 4, 5, d, 6 represet the first, the secod, the third, the fourth, the fifth, d the sixth terms, respectively. Fid the vlue of commo differece d usig the formul for clcultig the geerl term of rithmetic sequece. ( ) = + d = 6 = 0+ ( 6 )( d) 5= 0+( 5)( d) d = 5 5 d = 5 5 d = 3 Now, fid the vlues of, 3, 4, d ( )() = = +( )()= ( )() = = +( )()= ( )() = = +( )()= ( )() = = +( )()= Thus, the four rithmetic mes re -7, -4, -, d. The fiite rithmetic sequece is {-0, -7, -4, -,, 5}.