8.1Sequences A-level考试纯数学P1课件.pptVIP

Chapter8Sequences

ContentsConstructingsequences1Thetrianglenumbersequences2Thefactorialsequences3Pascalsequences4Arithmeticsequences5

Constructingsequences(a)1491625…(c)9997959391…(e)2481422…(b)…(d)11.11.211.3311.4641…(f)31415…Rowsofthiskindarecalledsequences,andtheseparatenumbersarecalledterms

Theusualnotationforthefirst,second,third,…termsofasequenceisu1,u2,u3,andsoonIfrisanaturalnumber,thentherthtermwillbeur

Writeaformulafortherthtermofthesequence(a)ur=r2(b)ur+1=ur–2u1=99and(c)ur+1=1.1uru1=1and(d)Definitionslikethesearecalledinductivedefinitions(e)u1=2andur+1=ur+2r

Whatabout(f)1,6,1…?9,2,6…?Asequencecanneverbeuniquelydefinedbygivingjustthefirstfewterms.Asequencecanonlybedescribedunambiguouslybygivingaformula,aninductivedefinitionintermsofageneralnaturalnumberr,orsomeothergeneralrule.

ThetrianglenumbersequenceThesumofallthenaturalnumbersfrom1torist1=1,t2=1+2=3,t3=1+2+3=6,andingeneraltr=1+2+3+…+r

++==2tr=(r+1)+(r+1)+(r+1)+…+(r+1)+(r+1)+(r+1)1+2+3+…+(r–2)+(r–1)+rr+(r–1)+(r–2)+…+3+2+1tr=tr=

Thefactorialsequencefr+1=fr×(r+1),wherer=0,1,2,3….f1=f0×1=1×1=1,f2=f1×2=1×2=2fr=1×2×3×…×r.(r≥1)Thissequenceissoimportantthatithasitsownspecialnotation,r!,readas‘factorialr’or‘rfactorial’(oroften,colloquially,as‘rshriek’)

ThefactorialsequenceFactorialrisdefinedby0!=1and(r+1)!=r!×(r+1),wherer=0,1,2,3,….Forr≥1，r!istheproductofallthenaturalnumbersfrom1tor.