EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 2- ALGEBRAIC TECHNIQUES TUTORIAL 1 - PROGRESSIONS
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1 EDEXCEL NATIONAL CERTIFICATE UNIT 8 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME - ALGEBRAIC TECHNIQUES TUTORIAL - PROGRESSIONS CONTENTS Be able to apply algebaic techiques Aithmetic pogessio (AP): fist tem (a), commo diffeece (d), th tem e.g. a +( )d ; aithmetic seies e.g. sum to tems, S a ) d Geometic pogessio (GP): fist tem (a), commo atio (), th tem e.g. a - ; a a geometic seies e.g. sum to tems, S, sum to ifiity S solutio of pactical poblems e.g. compoud iteest, age of speeds o a dillig machie Complex umbes: additio, subtactio, multiplicatio of a complex umbe i Catesia fom, vecto epesetatio of complex umbes, modulus ad agumet, pola epesetatio of complex umbes, multiplicatio ad divisio of complex umbes i pola fom, pola to Catesia fom ad vice vesa, use of calculato Statistical techiques: eview of measue of cetal tedecy, mea, stadad deviatio fo ugouped ad gouped data (equal itevals oly), vaiace It is assumed that the studet has completed the module MATHEMATICS FOR TECHNICIANS. D.J.Du
2 . ARITHMETIC PROGRESSION Coside the sequece of umbes 3,5,7,9,,. This is a pogessio that stats at 3 ad is iceased by each time so we say the commo diffeece is d = ad the statig value is a = 3. The sum of tems is called a ARITHMETIC SERIES. To fid this we would have to add them all up. The sum S would be: S a ( a d) a d d a d d d... a d d... a d a 3d ( a 4d)... a d d S a ( a d) It ca be show that the aswe is () x (aveage value). The aveage value is foud by addig the fist ad last tem ad dividig by so we ca wite: S a a d a S d a d S S a ) d This is the geeally accepted fomula fo evaluatig a pogessio. WORKED EXAMPLE No. Show that the fomula deived above gives the coect aswe fo the sum of the pogessio: Simply addig them up gives S = 5 Fo this pogessio = 5, d = ad a = S a ) d Usig the fomula gives WORKED EXAMPLE No. Evaluate the fist 0 tems of the pogessio a = 4 (the fist value) d = 4 (the diffeece betwee each umbe) = 0 Usig the fomula gives S a ) d 0 S S D.J.Du
3 WORKED EXAMPLE No. 3 A maufactue of steel bas oly sells them i batches of 0 m ad calculates the chage to customes usig the followig system. 0-0 m 0-0 m 0-30 m 00 ad so o with a maximum ode of 00 m Wite out the aithmetic pogessio ad the fomula fo calculatig the cost of a ode. Calculate the chage fo 00 m The pogessio is 0, 0, 00. The cost of batches is : S a ) d Fo 00 m = 0 a = 0 d = -0 0 S a ) d ()(0) 0 0 S The chage is 500 SELF ASSESSMENT EXERCISE No.. Evaluate the fist 5 tems of the pogessio Evaluate the fist 30 tems of the pogessio Evaluate the fist 0 tems of the pogessio How may tems ae equied i questio () to poduce a aswe of zeo? 5. Evaluate the fist 5 tems of the pogessio give i (). 6. A maufactue of a mass poduced poduct chages by the followig method. Quatity Pice ad so o with a maximum ode of 000 How much would a ode of 000 cost? (Aswe 390) (Aswe 570) (Aswe 00) (Aswe ) (Aswe 350) (Aswe 0) D.J.Du 3
4 . GEOMETRIC PROGRESSION A geometic pogessio is a seies of umbes statig at a ad the ext tem is the pevious tem multiplied by a commo atio. It follows that the pogessio is a, a, a, a 3..a If we add tems togethe we have a GEOMETRIC SERIES. Fo example let a =. The two gaphs show the affect of havig slightly lage ad slightly smalle tha. Whe is smalle tha the gaph shows that S eaches a fial value but whe is lage tha, S keeps gowig. The sum of tems would be give by: S = a + a + a + a 3.+ a S = a( ) multiply both sides by ( - ) ( - )S = a( ) ( - ) ( - )S = a{( ) - ( )} ( - )S = a{ - + )} a S Note this does ot wok if = Sometimes we wat to kow the value whe is ifiity. This is ot always possible but ofte the tems will geeally get smalle ad smalle ad covege o a value. This oly happes if the commo atio is less tha ad lage tha -. a Whe this happes the sum is S The poof is ot give hee. WORKED EXAMPLE No. 4 Fid the sum of the fist 6 tems of the geometic pogessio 5, 5(), 5(), 5() 3..5() a = 5, = ad = 6 6 a S a D.J.Du 4
5 WORKED EXAMPLE No. 5 Fid the sum of the fist 4 tems of the geometic pogessio 6, 6(3), 6(3), 6(3) 3..6(3) a = 6, = 3 ad = 4 4 a S a WORKED EXAMPLE No. 6 Fid the sum to ifiity of the seies, ( ½), (½ ), (½) 3..(½) a =, = ½ 4 a S a 4 / / SELF ASSESSMENT EXERCISE No.. Evaluate the fist 0 tems of the seies 3 + 3x + 3x + 3x 3 whe x = (Aswe 64). Evaluate the maximum value of the seies 0 + 0x + 0x + 0x 3 whe x = ½ (Aswe 40) 3. Evaluate the maximum value of the seies 0 + 0x + 0x + 0x 3 whe x = -½ (Aswe 3.333) D.J.Du 5
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