Summation Notation The sum of the first n terms of a sequence is represented by the summation notation i the index of summation

Transcription

1 Lesso 0.: Sequeces d Summtio Nottio Def. of Sequece A ifiite sequece is fuctio whose domi is the set of positive rel itegers (turl umers). The fuctio vlues or terms of the sequece re represeted y, 2, 3,...,.... Sequeces whose domis cosist oly of the first terms re clled fiite sequeces. O TI-83 clcultor sequece is foud: seq(geerl term, x,strtig term, edig term, step) Fctoril Nottio If is positive iteger, the ottio!, red fctoril, is the product of ll positive itegers from dow through.! ! y defiitio Summtio Nottio The sum of the first terms of sequece is represeted y the summtio ottio i the idex of summtio i... where i 2 3 the upper limit of summtio the lower limit of summtio Note:. The lower limit of summtio c e y umer. 2. Writig out sum tht is give i summtio ottio is clled expdig the summtio ottio. O TI-83 clcultor the sum of sequece is foud: sum(seq(geerl term, x,strtig term, edig term, step))

2 Lesso 0.2: Arithmetic Sequeces Def. of Arithmetic Sequece A rithmetic sequece is sequece i which ech term fter the first term differs from the precedig term y costt mout. The differece etwee cosecutive terms is clled the commo differece, d, d is foud y sutrctig 2 successive terms, for exmple 2. Geerl Term of Arithmetic Sequece The th term (the geerl term) of rithmetic sequece with first term d commo differece, d, is ( ) d. Exmple: Accordig to the U.S. Bureu of Ecoomics Alysis, U.S. trvelers spet $2,808 millio i other coutries i 984. This mout hs icresed y $2350 millio yerly.. Write formul for the th term, yers fter How much will U.S. trvelers sped i 200? The Sum of the First Terms of Arithmetic Sequece The sum, S, of the first terms of rithmetic sequece is give y S 2 where is the first term d is the th term.

3 Lesso 0.3: Geometric Sequeces Def of Geometric Sequece A geometric sequece is sequece i which ech term fter the first term is otied y multiplyig the precedig term y fixed o-zero costt. The mout y which we multiply ech time is clled the commo rtio, r, of the sequece. Geerl Term of Geometric Sequece The th term (the geerl term) of geometric sequece with first term,, d commo rtio, r, is r The Sum of the First Terms of Geometric Sequece The sum, S, of the first terms of the geometric sequece is give y ( r S ) r which is the first term d r is the commo rtio r. i Exmple: A jo pys slry of $30,000 the first yer. Durig the ext 29 yers, the slry icreses y 6% ech yer. Wht is the totl lifetime slry over the 30 yer period? The Sum of Ifiite Geometric Series If r is i the itervl,, the the sum of the ifiite geometric series 2 3 r r r... i which is the first term d r is the commo rtio is give y S. If r is ot i the itervl,, the ifiite series does ot hve sum. r

4 Lesso 0.4: Mthemticl Iductio Needed to e updted The Steps i Proof y Mthemticl Iductio Let S e sttemet ivolvig the positive iteger. To prove tht S is true for ll positive itegers requires two steps: Step : Show tht S is true. Step 2: Show tht if S k is ssumed to e true, the S k is lso true, for every positive iteger k. This is usully doe y ddig k to oth sides of S k, d showig tht the result is S k.

5 Lesso 0.5: The Biomil Theorem Expdig ( ) : Ptters:. The first term i. The expoet of decreses y i ech successive term. 2. The expoet of icreses y i ech successive term. I the first term, the expoet of is 0. The lst term is. 3. The sum of the expoets o the vriles i y term is equl to, the expoet of ( ). 4. There is oe more term i the polyomil expsio th there is i the power of the iomil. 5. The coefficiet of ech term fter the first term is sed upo the precedig term. It is the coefficiet of the precedig term times the power of of the precedig term divided y the umer of precedig term. Pscl s Trigle: The first d lst umers i ech row re. Ech of the other umers is otied y fidig the sum of the closest two umers ove it.

6 Lesso 0.5: The Biomil Theorem, pge 2 Defiitio of Biomil Coefficiet For oegtive itegers d r with r, the expressio! clled iomil coefficiet d is defied y. r r!( r)! r r (red ove r ) is The Biomil Theorem For y positive iteger, Fidig Prticulr Term i Biomil Expsio The rth term of the expsio of is r r r

7 Lesso 0.6 Coutig Priciple, Permuttios, d Comitios The Fudmetl Coutig Priciple The umer of wys i which series of successive thigs c occur is foud my multiplyig the umer of wys ech thig c occur. Exmple: A pizz hs 3 choices for size, 4 choices for crust, d 6 choices for toppigs. How my differet oe-toppig pizzs c e ordered? Exmple: A multiple choice test with 6 questios d 3 choices for ech questio c e swered how my wys? Exmple: A licese plte i prticulr stte displys 2 letters followed y 3 umers, such s CB-23. How my differet licese pltes c e mufctured? Permuttios A permuttio is ordered rrgemet of items tht occur whe. No item is used more th oce. 2. The order of the rrgemet mkes differece. Exmple: Assume there re 2 girls o softll tem. How my wys c the ttig order e set up? Permuttios of Thigs Tke r t Time The umer of possile permuttios if r thigs re tke from items is P r! ( r)! The ottio P r mes the umer of permuttios of thigs tke r t time.

8 Lesso 0.6 Coutig Priciple, Permuttios, d Comitios, pge 2 Exmple: A corportio hs seve memers o its ord of directors. I how my differet wys c it elect presidet, vice-presidet, secretry d tresurer? Exmple: I how my wys c 6 ooks e lied up log shelf? Exmple: O our clcultor, we hve P r key uder mth, proility. Let clculte 7P 4. Comitios A comitio of items occurs whe:. Items re selected from the sme group. 2. No item is used more th oce. 3. The order of the items mkes o differece. Comitios of Thigs Tke r t Time The umer of possile comitios if r items re tke from items is C r! ( r)! r! The ottio C r mes the umer of comitios of thigs tke r t time. Exmple: From group of 0 physicis, i how my wys c four people e selected to tted coferece? Exmple: How my differet 4-crd hds c e delt from deck tht hs 6 differet crds?

9 Lesso 0.7: Proility Empiricl Proility Empiricl Proility pplies to situtios i which we oserve how frequetly eve ctully occurs. Computig Empiricl Proility The empiricl proility of evet E is PE ( ) the oserved umer of times E occurs the totl umer of oserved occureces Exmple: Usig the chrt of pge 984, fid the proility of rdomly selectig Americ who gets 7 hours of sleep o typicl ight. Theoreticl Proility A experimet is y occurrece for which the outcome i ucerti. The smple spce of experimet, deoted S, is the set of ll possile eqully likely outcomes of experimet. Computig Theoreticl Proility If evet E hs E ( ) eqully likely outcomes d its smple spce hs S ( ) eqully likely outcomes, the theoreticl proility of evet E, deoted PE ( ), is PE ( ) the umer of times E occurs E ( ) the umer i the smple spce,s S ( ) The sum the theoreticl proilities of ll possile outcomes i the smple is lwys. Exmple: A die is rolled. Fid the proility of gettig umer less th 5. Exmple: Wht is the proility of gettig sum of 5 whe two six-sided dice re rolled?

10 Lesso 0.7: Proility, pge 2 Exmple: You re delt crd from stdrd 52-crd deck. Fid the proility of eig delt hert. Exmple: I lottery ssume plyer chooses 5 differet umers from to 30. Wht is the proility of mtchig the 5 wiig umers? The Proility of Evet Not Occurrig The proility tht evet E will ot occur is equl to oe mius the proility tht it will occur. P( ot E) P( E ) Exmple: Bsed upo the previous exmple, with oe lottery ticket, wht is the proility of ot wiig? Mutully Exclusive Two evets, A d B, re mutully exclusive if it is impossile for A d B to occur simulteously. Or Proilities with Mutully Exclusive Evets If A d B re mutully exclusive evets, the P( A or B) P( A) P( B ) Exmple: If you roll six-sided die, wht is the proility of gettig either 4 or 5? Exmple: If oe crd is rdomly selected from deck of crds, wht is the proility of selectig Kig or Quee?

11 Lesso 0.7: Proility, pge 3 Or Proilities with Evets tht re NOT Mutully Exclusive If A d B re ot mutully exclusive evets, the proility of A or B is the proility of A plus the proility of B mius the proility tht A d B occur simulteously. P( A or B) P( A) P( B) P( A d B ) Exmple: I group of 25 oos, 8 ejoy pickig fles off their eighors d 6 ejoy screechig wildly. These figures iclude 0 oos tht ejoy oth ctivities (pickig fles d screechig wildly). If oe oo is selected t rdom from the group, fid the proility tht it ejoys pickig fles or screechig wildly. Idepedet Evets Two evets re idepedet if the occurrece of oe of them hs o effect o the proility of the other. Ad Proilities d Idepedet Evets If A d B re idepedet evets, the the proility of A d B is the proility of A times the proility of B. P( A d B) P( A) P( B ) Exmple: A roulette wheel hs 38 comprtmets. Eightee re lck, 8 re red d 2 re gree. Fid the proility of ldig o red o 2 cosecutive turs. Exmple: Fid the proility of fmily hvig 5 girls i row.