8.2Sequences A-level考试纯数学P1课件.ppt

Sequences

Chapter8

Pascalsequencesandwherer=0,1,2,3,…AtypicalexampleofPascalsequencesWritethetermsp1,p2,…p6

Usingtheinductivedefinitionforr=0,1,2,…inturnproducesthetermsThecompletesequenceis1,4,6,4,1,0,0,0,0,0,ThisisonlyoneofafamilyofPascalsequences

NotationofPascalsequencesForexampleOtherPascalsequenceshavenumbersdifferentfrom4inthemultiplyingfactor

ThegeneraldefinitionofPascalsequences,whosetermsaredenotedbyisandwherer=0,1,2,3,…

133114641121111133114641121111ThecompletepatternofPascalsequences,withoutthetrailingzeros,iscalledPascal’striangleEvernumberinthepatternexceptforthe1sisthesumofthetwonumbersmostcloselyaboveit.

ArithmeticsequencesTheinductivedefinitionforanarithmeticsequencehastheformu1=a,ur+1=ur+dthenumberdiscalledthecommondifference.

ExampleSennewouldliketogiveasumofmoneytoacharityeachyearfor10years.Shedecidestogive﹩100inthefirstyear,andtoincreasehercontributionby﹩20eachyear.Howmuchdoesshegiveinthelastyear,andhowmuchdoesthecharityreceivefromheraltogether?

Theyusuallyonlycontinueforafinitenumberofterms.Itisofteninterestingtoknowthesumofalltheterms.Inthiscase,itisusualtodescribethesequenceasaseries.

a,a+d,a+2d,a+3d,…Anarithmeticseriesofntermswithfirsttermaandcommondifferencedhaslastterml=a+(n–1)dandsum

ExampleFindthesumofthefirstnoddnaturalnumbers

ExampleAstudentreadinga426bookfindsthathereadsfasterashegetsintothesubject.Hereads19pagesonthefistday,andhisrateofreadingthengoesupby3pageseachday.Howlongdoeshetaketofinishthebook?