,... are the terms of the sequence. If the domain consists of the first n positive integers only, the sequence is a finite sequence.

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1 Chpter 9 & 0 FITZGERALD MAT 50/5 SECTION 9. Sequece Defiitio A ifiite sequece is fuctio whose domi is the set of positive itegers. The fuctio vlues,,, 4,...,,... re the terms of the sequece. If the domi cosists of the first positive itegers oly, the sequece is fiite sequece. ( x) x 5 f is the sme s 5 Fid the first five terms of 5. Fid the first five terms of ( ). Fidig sequece fuctio.,, 5, 7,., -5, 0, -7,

2 Chpter 9 & 0 FITZGERALD MAT 50/5 Recursive sequece is sequece fuctio where the ext term i the sequece is determied by the vlues of the previous terms d ot the vlue of.,, 0 where. Fctoril Nottio If is positive iteger, fctoril is defied s ( " ) ( " ) As specil cses, zero fctoril is defied s 0 d Write the first five terms give by Begi with 0.

3 Chpter 9 & 0 FITZGERALD MAT 50/5 Simplify the fctorils by hd ( ) Summtio Nottio. The sum of the first terms of sequece is represeted s " 4... Where is clled the idex of summtio, is the upper limit of summtio d is the lower limit of summtio. Fid ech sum. 5 6 ( ) Write i summtio ottio. 9 """ 4 8

4 Chpter 9 & 0 FITZGERALD MAT 50/5 4 Properties of Sums c is costt.. c c. c c. ( ) b b 4. ( ) " " b b 5. " j j 6. ( ) " 7. ( )( ) 6 " 8. ( ) " # $ % & ''' ( Fid ech sum. 0 ( ) 8 ( ) " 4 7 ( ) 0 6 4

5 Chpter 9 & 0 FITZGERALD MAT 50/5 5 SECTION 9. A sequece is rithmetic if the differeces betwee cosecutive terms re the sme. The sequece,,,...,,... is rithmetic if there is umber d such tht d, c be defied s d( ). Fid the 40 th term of the sequece, 6, 0, 4, 8. If so, the rithmetic sequece equtio Fid the rithmetic sequece fuctio if the 4 th term is 0 d the th term is 65. Arithmetic Series Fid the sum of the first 00 positive itegers ? Fid the 50th prtil sum of 5, 6, 7, 8, 49 Fid the sum of

6 Chpter 9 & 0 FITZGERALD MAT 50/5 6 SECTION 9. A sequece is geometric if the commo rtios betwee cosecutive terms re the sme. The sequece,,,...,,... is geometric if there is umber r such tht r 4... ( ) " r., 4 Fid the 0 th term of the sequece, 6, 8, 54. If so, the geometric sequece equtio c be defied s Fid the geometric sequece fuctio if the rd term is 6 d the 5 th term is6. Geometric Series Fid the formul. S r r """ r ( ) Fid the 0 th prtil sum of 5, 0, 0, 40

7 Chpter 9 & 0 FITZGERALD MAT 50/5 7 Fid the ifiite sum of Prove tht the repetig deciml equls. Determie whe series diverge or coverge.

8 Chpter 9 & 0 FITZGERALD MAT 50/5 8 SECTION 0. Subsets d Coutig Priciples. If every elemet i set A is lso elemet of set B, we sy tht A is subset of B d write A B. If A B d A B, we sy tht A is proper subset of B d write A B. We should lso gree tht the empty set, or { }, is subset of every set. Write dow ll the possible subsets of the set {, b, c}. Fidig the umber of subsets d the crdil umber of set. Defie Uio d Itersectio of sets. I survey of 00 college studets, 5 were registered i College Algebr, 5 were registered i Eglish, d 8 were registered i both courses. How my were registered i College Algebr or Eglish? How my were registered i either clss?

9 Chpter 9 & 0 FITZGERALD MAT 50/5 9 Me tree digrm for two ppetizers, 4 etrees, d desserts. Coutig Priciple. The Studet Uio is hvig luch specil vlue mel. You get to choose oe of 4 sdwiches, oe of 5 bgs of chips, oe of 7 dris d oe of desserts. How my differet luch specils c you me? SECTION 0. Permuttios d Combitios. A permuttio is ordered rrgemet of r objects chose from objects. Type. Distict Objects with Repetitio. The umber of ordered rrgemets of r objects chose from objects, i which the objects re distict d repetitio is r llowed, is. How my letter Airport bbrevitios exist whe the letters re llowed to repet? Type. Distict Objects without Repetitio. The umber of ordered rrgemets of r objects chose from objects, i which the objects re distict d repetitio is ot llowed, is P(, r) Pr ; r. ( " r) How my letter Airport bbrevitios exist whe the letters re ot llowed to repet?

10 Chpter 9 & 0 FITZGERALD MAT 50/5 0 A combitio is groupig, rrgemet where order does t mtter, of r objects selected from distict objects without repetitio, where r <. The ottio is ) & C(, r) ' $ Cr ; r ( r % ( # r) " r Give the first 5 letters of the lphbet, how my letter groups re there? How my perso committees of people c be formed from 8 people? How my groups of 4 people, me d wome, if there re 5 me d 6 wome to choose from? Distict Permuttios. The umber of permuttios o objects of which r re of oe id, r re of secod id, r re of third id,, r re of th id is give by C(, r )" C( r, r )" C( r r, r )"..." C( r r... r, r ) or this simplifies to where r r r r r r r... r... How my distict letter rrgemets re there for the word MISSISSIPPI?

11 Chpter 9 & 0 FITZGERALD MAT 50/5 SECTION 9.5 Pscl s Trigle d Biomil Expsio. How to use the grphig clcultor to fid row of Pscl s trigle Biomil Expsio. Expd ( x ) 4

12 Chpter 9 & 0 FITZGERALD MAT 50/5 7 Fid the coefficiet of x of ( x ) 0. Fid the 7 th term of ( x ) Probbility. Ay hppeig for which the result is ucerti is clled experimet. The possible results of the experimet re outcomes, the set of ll possible outcomes of the experimet is the smple spce of the experimet, d y sub-collectio of smple spce is evet. Fidig Smple spce. Tree Digrm, Mtrix, Coutig Priciples. Smple spce for Flippig coi three times. Rollig two die. A dec of crds.

13 Chpter 9 & 0 FITZGERALD MAT 50/5 Probbility will lwys be the umber of outcomes of the desired evet out of the totl umber of outcomes i the smple spce. P(Evet) will hve the followig rge Fid the followig probbilities 0 < P(E) < P( tils i flips) P(t lest tils i flips) P(rollig 7) P(rollig vlue > 8) P(fce crd) Probbility with d d or. P(A d B) re the outcomes tht stisfy both A d B. P ( A " B) P( A) P( B) P(A or B) re the outcomes tht stisfy A or B or both. P ( A # B) P( A) P( B) " P( A B) P(Herts d Fce Crds) P(Herts or Fce Crds)

14 Chpter 9 & 0 FITZGERALD MAT 50/5 4 Complemet Probbility. Let A be evet d let A be its complemet. P(A) P( A ) or P( A ) P(A). P(t lest oe tils i 5 flips) I group of 0 people, wht is the probbility tht t lest people hve the sme birthdy? Biomil Probbility. This probbility hs oly two outcomes, success or filure. Biomil Probbility Formul The spirit club is 75% girls. Wht is the probbility of rdomly picig 5 people, where there re boys d girls? A mufcture hs determied tht bulb mchie will produce bd bulb for every 000 bulbs it produces. Wht is the probbility tht order of 00 bulbs is ll perfect? There is t lest oe bd bulb? There is oe bd bulb? There re two bd bulbs?

15 Chpter 9 & 0 FITZGERALD MAT 50/5 5 Probbilities of ctul experimets I bg of M & Ms, the cdies re colored red, gree, blue, brow, yellow, d orge. Sy we ope bg d couted ll the colors to get the followig tble. Color Number Probbility Red 6 Gree Blue Brow Yellow 5 Orge Probbility with & without replcemet. Arrgemets vs Groups. A bg cotis 5 red chips, white chips, d blue chips. Fid the followig probbilities. P( red, white, d blue i order with replcemet) P( red, white, d blue i order without replcemet) P( oe of ech color with replcemet) P( oe of ech color without replcemet)

is an ordered list of numbers. Each number in a sequence is a term of a sequence. n-1 term

is an ordered list of numbers. Each number in a sequence is a term of a sequence. n-1 term Mthemticl Ptters. Arithmetic Sequeces. Arithmetic Series. To idetify mthemticl ptters foud sequece. To use formul to fid the th term of sequece. To defie, idetify, d pply rithmetic sequeces. To defie rithmetic