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1、Copyright©byHoughtonMifflinCompany,Inc.Allrightsreserved.1PatternsandSequencesPatternsandSequencesPatternsrefertousualtypesofproceduresorrulesthatcanbefollowed.Patternsareusefultopredictwhatcamebeforeorwhatmightcomeafterasetanumbersthatarearrangedinaparticularorder.Thisarrangementofnum
2、bersiscalledasequence.Forexample:3,6,9,12and15arenumbersthatformapatterncalledasequenceEachnumberinasequenceiscalledterms.PatternsandSequencesAfinitesequenceendsafteracertainnumberofterms.Aninfinitesequenceisonethatcontinuesindefinitely.ArithmeticSequencesandSeriesDigitalLessonCopyrigh
3、t©byHoughtonMifflinCompany,Inc.Allrightsreserved.5Aninfinitesequenceisafunctionwhosedomainisthesetofpositiveintegers.a1,a2,a3,a4,...,an,...DefinitionofSequenceThefirstthreetermsofthesequencean=4n–7area1=4(1)–7=–3a2=4(2)–7=1a3=4(3)–7=5.finitesequencetermsCopyright©byHoughtonMifflinCompa
4、ny,Inc.Allrightsreserved.6Asequenceisarithmeticifthedifferencesbetweenconsecutivetermsarethesame.4,9,14,19,24,...DefinitionofArithmeticSequence9–4=514–9=519–14=524–19=5arithmeticsequenceThecommondifference,d,is5.Copyright©byHoughtonMifflinCompany,Inc.Allrightsreserved.7Example:Findthef
5、irstfivetermsofthesequenceanddetermineifitisarithmetic.an=1+(n–1)4Example:ArithmeticSequenceThisisanarithmeticsequence.d=4a1=1+(1–1)4=1+0=1a2=1+(2–1)4=1+4=5a3=1+(3–1)4=1+8=9a4=1+(4–1)4=1+12=13a5=1+(5–1)4=1+16=17Copyright©byHoughtonMifflinCompany,Inc.Allrightsreserved.8Thenthtermofanari
6、thmeticsequencehastheforman=dn+cwheredisthecommondifferenceandc=a1–d.ThenthTermofanArithmeticSequence2,8,14,20,26,....d=8–2=6a1=2c=2–6=–4Thenthtermis6n–4.Copyright©byHoughtonMifflinCompany,Inc.Allrightsreserved.9a1–d=Example:FormulaforthenthTermExample:Findtheformulaforthenthtermofanar
7、ithmeticsequencewhosecommondifferenceis4andwhosefirsttermis15.Findthefirstfivetermsofthesequence.an=dn+c=4n+1115,d=4a1=1519,23,27,31.Thefirstfivetermsare15–4=11Copyright©byHoughtonMifflinCompany,Inc.Allrightsreserved.10TheSumofaFiniteArithmeticSequenceThesumofafinitearithmeticsequenc
