Jso Mille 8 Udestd the piciples, popeties, d techiques elted to sequece, seies, summtio, d coutig sttegies d thei pplictios to polem solvig. Polomil Diffeece Theoem: f is polomil fuctio of degee iff fo set of -vlues tht foms ithmetic sequece, the th diffeeces of coespodig -vlues e equl d o-zeo. Emple: Coside the fuctio Y fo. 5... 6 8 Sutctig the cosecutive vlues we see tht the diffeeces e equl 6 8 Sice is lie fuctio fist degee we ol hd to tke the st diffeeces. Howeve, s the degee of the polomil iceses, the diffeeces we must tke to get to costt must equl to the degee of the polomil. Emple: Coside the fuctio 5 6 8 6 96 7 st diffeece 8 5 78 d diffeece 6 8 d diffeece 6 6 6 I the cse of thid degee polomil we took the thid diffeece d foud them to e costts thus the polomil diffeece holds. Fiite Diffeeces: Tke tle of -vlues coespodig to ithmetic sequece of -vlues, d compute the diffeeces of cosecutive -vlues. If those vlues of those diffeeces evetull ecome costt the the fuctio is polomil of degee equl to the ume of diffeeces tke to ech the costt.
Sequeces d Seies Aithmetic Lie Sequece: A sequece with costt diffeece, meig tht ech successive tem is lge tht the oe pecedig it. Tht is: >, d, o d, whee d is costt. Emple: Fo the sequece, 6, 9,, 5, 8, d d 6. The eplicit fomul is: -*. Aithmetic Seies: A sum of tems with commo diffeece d. The sum of the fist tems, epeseted t, is t. t o t [ d] Geometic Epoetil Sequece: A sequece with costt tio etwee tems fomed sttig with ume d multiplig ech successive tem costt. Tht is: > g * g, o whee is o-zeo costt. Emple: Fo the sequece,, 5, 5, 5, 5, d 5. The eplicit fomul is: 5 Geometic Seies: the sum of cosecutive tems of geometic sequece. This c e i foud with the followig fomul: iteges. i Fiocci Sequece: 5 8 55 89.. Nmed fte Leodo of Pis k Fiocci who studied the sequece d its popeties i the th cetu. It is foud i supisig ume of loctios i tue icludig piecoes, su flowes, disies, d eve sil shells Ech successive tem is foud ddig the two tems efoe it. The eplicit fomul to clculte the th tem 5 5 is 5 Howeve, the followig ecusive fomul is esie to use d is moe t the high school level., fo
Pemuttios d Comitios Pemuttio: igmet of ojects i specific ode. The ume of pemuttios of ojects is deoted! which is defied s:! Emple: ABC c e pemutted 6 ws : ABC, ACB, CBA, CAB, BAC, BCA! 6 Fudmetl Coutig Piciple: If thee e ws to do oe thig, d m ws to do othe, the thee e m* ws of doig oth. Emple: Suppose ou oll die d flip coi. Thee e outcomes fo flippig coi d 6 outcomes of ollig dice. B the Fudmetl Coutig Piciple thee e *6 ws of doig oth. Pemuttio of ojects tke t time: The ume of pemuttios of ojects! tke t time, deoted P,, o P whee.! Emple: Fid the ume of ws to ed diffeet ooks selected fom collectio of.!! 9! P! 9! 9! Comitio: A igmet of ojects i which ode does ot mtte.! Comitio of ojects tke t time:!! Emple: I collectio of ooks thee e 8 msteies d omce ovels. How m ws e thee to choose msteies d omce ovels, espectivel? 8 8! Selectig msteies 56!8!! Selectig omces 6!! Usig the fudmetl coutig piciple 56*6 6 possile ws.
Pscl s Tigle d Biomil Coefficiets The iomil epds to. If we look t iomils with othe epoetil powes we develop the followig ptte i the shpe of tigle. The fist d lst eties i ech ow e lws d ll othe eties e foud ddig the tems foud immeditel ove to the left d of it. It is esult of the epsio of. Emples: Pscl s tigle is es to use epesettio of the epsio of, d hs m popeties ll the powes of fom to occu i ode I ech tem, the epoets of d dd to If the powe of is, the the coefficiet of the tem is The th ow cotis eties. 5 The k th et i ow is C k- 6 The sum of ll the eties i ow equls Biomil Theoem: Emple: Epd 7 6 o
Polems:. Fom goup of smokes d 5 osmokes, uivesit eseche will choose 5 idividuls to pticipte i stud o lug cce.. How m ws c this e doe if it does ot mtte how m smoke o osmokes e icluded?. I how m ws c this e doe if ectl smokes must e chose? c. Fid the poilit tht ectl of the 5 people doml selected fom the goup of smokes.. Geometic figues e ofte epeseted the lettes tht idetif ech vete. Detemie the ume of ws tht ech figue elow c e med with the lettes A, B, C, D, E, d F. Assume tht ech figue is iegul.. Tigle. Qudiltel c. Which two shpes will hve the sme ume of possile mes?. I the figue elow, the sum of ech digol is t the ed of the digol.. Fid the missig sums.. Wht is the me of the sequece fomed the sums? Schultz, Ellis, Hollowell, Keed Wisto,, pp. 79???? 6. 5 5 6 5 5 6 7 5 5 7.. How m sques e o 8 8 checkeod?. Let f the ume of sques o checkeod. Is f polomil fuctio? Justif ou swe.
5. Thee is sto ofte told out Kl Fiedich Guss. At ge he poved the Fudmetl Theoem of Alge. Whe he ws i thid gde, his clss misehved d the teche gve the followig polem s puishmet: Add the iteges fom to. It is sid tht Guss solved the polem i lmost o time t ll lettig S e the desied sum, usig the commuttive d ssocitive popeties to wite the sum i evese ode, d ddig the coespodig tems. The sums wee ll the sme, s, so S. Detemie wht is d wht Kl detemied the sum to e. Sek, 99, pp. 7. 6. Suppose thete hs 6 sets i the fist ow d tht ech ow hs moe sets th the pevious ow. If thee e ows i the thete,. how m sets e i the lst ow?. how m sets e thee i ll? Sek, 99, pp. 75 7. Wite the seies usig Σ-ottio:. 6 8. 9 8 6 7 88 576 5 c. the sum of the sques of the iteges fom to Sek, 99, pp. 757 8. A m stts with pi of its. How m pis of its c e poduced fom tht fist pi if the give ith to ew pi ech moth d it tkes moth fo pi to mtue d epoduce? LIVIO,
Aswes:.. 5 C 5 5,. C 5 C * 5 6 c.6/5,.7%! 6!.. P!!. 6 c. Petgo d Hego.. The missig umes e 5 8 d if the tigle ws cotiued 55 89. This is the Fiocci sequece. Cout the ume of sques, up to 88 sques. dimesio of side of od: 5 6 7 8 f # of sques: 5 55 9. Yes f is d degee polomil ecuse the thid diffeeces e costt. f 6 5. Guss let S 9899. Usig the Commuttive d Associtive Popeties, the sum c e witte i evese ode: S9998. Addig the two togethe we get S times S* S55 So is d Kl detemied the sum to e 55. 6.. To fid out how m sets e i the lst ow multipl the ume of ows the ume of sets eig dded ech ow sutctig tht e ot dded to the fist ow: *-. To fid the totl ume of sets i the thete usig the fomul: t [ d] t t t 5 5 [ 6 ] [ 68] 7 7.. i. 9 i 8 i i c. i i 8. Moth 5 6 7 8 9 Pis 5 8 55 89
Refeeces Livio, M.. The Golde Rtio. New Yok: Rdom House. Pickove, C.A. 5. A Pssio fo Mthemtics. Hooke, NJ: Joh Wile & Sos Schultz, J.E., Ellis, W., Hollowell, K.A., & Keed, P.A.. Alge. New Yok: Holt, Rieht, Wisto. Sek, S. L., Et Al. 99. Advced Alge. Illiois: Scott Foesm. Wlpole, R.E., Mes, R.H., Mes, S. L., & Ye, K.. Poilit d Sttistics fo Egiees & Scietists. 7 th ed. Uppe Sddle Rive, NJ: Petice Hll.