Chuyên đề nhị thức newton và ứng dụng.pdf

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Chuynnhthcnewtonvngdng21,005views

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1.Tinhiutiliu,thihntibookbooming.comCHUYNNHTHCNEWTONVNGDNG(NguynTrungHiu)A.LTHUYT:1.Cchngngthc(a+b)=10(a+b)=a+b1(a+b)=a2+2ab+b22(a+b)=a3+3a2b+3ab2+b33(a+b)=a4+4a3b+6a2b2+4ab3+b44...2.NhthcNewton(Niutn)a.nhl:n(a+b)=Cnan+Cnan1b+...+Cn1abn1+Cnbn=nankbkn01nnCkk=0Ktqu:nkn*(ab)=a+(b)=nank(b)=1)Cnankbk(nnkCkkk=0k=0n*(1+x)=n.xk=Cn+Cn.x+...+Cn.xnnCk01nk=0b.TnhchtcacngthcnhthcNiutn(a+b):nSccshngcacngthcln+1Tngsmcaavbtrongmishnglunlunbngsmcanhthc:(nk)+k=nShngtngqutcanhthcl:T=Cabk+1knnkk(lshngthk+1trongkhaitrin(a+b))nCchsnhthccchuhaishngu,cuithbngnhau.2=C+C+...+Cnnnnn10n0=CC+...+()C01nnn1nnTamgicpascal:1Khivitcchslnltvin=0,1,2,...tacbngnk2.Tinhiutiliu,thihntibookbooming.com012345....0111121213131414641515101051Trongtamgicsny,btuthngthhai,mishngthntctthhainctn1bngtnghaisnghngtrncngctvcttrcn.SdcquanhnyldoccngthctruyhiC=C+Ckn(Vi1



(HBKHN2000)Giibtphngtrnh:2A2xAxCx+102xGii:iukin:xlsnguyndngvx3Tac:dtphngtrnhchotngngvi:(2x1)2x6(x2)(x1)(x1)x+1023!x2x(2x1)x(x2)(x2)(x1)+103x12x4Vxlnghimnguyndngvx3nnx{34}Vd3:(HKA2004)Tmhscax8trongkhaitrinathcca:1+x2(1x)8Gii:kk8k8f(x)=C8x2(1x)=C8x2k(1)Ckxi.iCch1:Tac:kkik=0k=0i=04.Tinhiutiliu,thihntibookbooming.comi=00ik8k=4Vytachscax8l:(1)tho2k+i=8iC8kCkii,ki=2k=3Hstrongkhaitrincax8l:()C8C4+()C8C3=23804032211Cch2:Tac:(x)=C8++C8x2(1x)+C8x2(1x)++C8x2(1x)0343848f......Nhnthy:x8chctrongccshng:C8x2(1x)3Shngth4:3C8x2(1x)4Shngth5:4Vihstngngvi:A8=C8C3+C8C43240=238Vd4:(HHCQG,2000)121a)Tmhsx8trongkhaitrin1+xb)Chobittngttccchscakhaitrinnhthc(x+1)n2bng1024.Hytmhsa(a*)cashngax12trongkhaitrin.(HSPHN,khiD,2000)Gii:a)Shngth(k+1)trongkhaitrinl:k1(0k12)ak=C12x12x=C12x122kkkxTachn12k=228k=Vyshngth3trongkhaitrinchax8vchsl:C12=662nb)Tac:(1+x2)=Cnkx2n=Cn+Cnx2+...+Cnx122kk1kk=0Vix=1th:2n=n+n++n=C0C1...Cn1024n=10=22n10Dohsa(cax12)l:C=210610Vd5:(HVKTQS,2000)Khaitrinathc:P(x)=+x)12=a0+1x++12x12(12a...aTmmax(a0,a1,a2,...,a12)Gii:Giaklhslnnhtcakhaitrinsuyra:ak>ak1Tytachphngtrnh:5.Tinhiutiliu,thihntibookbooming.com212kC122k1C121kkk12k+1kkk+1k+12C122C121212kk+1ax(a0,a1,a2,...,a12)=a8=12218=mC81267202.Bitontmshngtrongkhaitrinnewton.Vd6:Tmshngth21trongkhaitrin:(23x)25Gii:Shngth21trongkhaitrinl:(x)20C2025253=C2525320x2020Vd7:a.Tmshngnggiatrongcckhaitrinsau(x+xy)21320x4x+1b.Tmshngnggiatrongcckhaitrinsau(xy)32Gii:a.Khaitrin(x+xy)203c21+1=22shngnnchaishngnggialsth11v12.C21(x3)11Shngth11l:(xy)1010=C21x43y1010C21(x3)10Shngth12l:(xy)1111=C21x41y111020x4x+1b.Khaitrinc20+1=21shng.Nnshngnggia2sls(xy)23101010476520212hngth2+1=16:C20x(xy)=C20xy10633(Vi[x]lkhiuphnnguyncaxnghalsnguynlnnhtkhngvtqux).Vd8:(HKhiD2004)Tmshngkhngchaxtrongkhaitrin.71f(x)=3x+4vix>0xGii:k77()17kkShngtngquttrongkhaitrin:Tk+1=C7k3x4=C7xk312(k,k7)x77ngvishngkhngchaxtac:k=0k=4312Vyshngkhngchaxtrongkhaitrinf(x)l:C7=3546.Tinhiutiliu,thihntibookbooming.comVd9:(HSPHN2001)Chokhaitrinnhthc:1012+x=a0+a1x+...+a9x+a10x.91033Hytmshngaklnnht.Gii:101211n1x=10(1+2x)=10C(2x)10kTac:+k10ak=C102kk3333k=0310akak+1C102kC10+12k+1kkkkakak1k1k1C102C1022k10!2k10!12k!10k!k+1!9k!Tacaktcmax()()()10kk+11922k2233kk210!210!k!(10k)!(k1)!(11k)!k11kk=7(k,k[0,10])277Vymaxak=a7=C10310BitppdngBi1:(HTK2002)Gia1,a2,,a11lcchstrongkhaitrinsau:(x+)(x+2)=x11+a1x101++11...aHytmhsa5Bi2:Tmhscax5trongkhaitrinx(12x)+x2(1+3x)(KhiD2007)510Bi3:Tmhscax5y3z6t6trongkhaitrinathc(x+y+z+t)(4TH&TT202003)Bi4:(TTHchuynPhanBiChuNghAn)Xcnhhscax11trongkhaitrin(x+2)(3x+1)nnathc:23bit:C2nC2n++)3kC2n++32nC2n=32n1...(12n2nkk0...1024n1Bi5:(LAISAC)KhaitrinP(x)=x3+2tac2xP(x)=a0x3n+1x3n+2x3n+a5a10...Bitrngbahsua0,a1,a2lpthnhcps4cng.TnhshngthxII.pdngnhthNewtonchngminhhthcvtnhtngthp.1.ThunnhthcNewtonDuhiunhnbit:KhiccshngcatngcdngCnankbkkthtas7.Tinhiutiliu,thihntibookbooming.comn(a+b)=CnankbknkdngtrctipnhthcNewton:.Viccnlichlk=0kholochna,b.Vd10:Tnhtng316C1615C16+14C16+1603132...C16Gii:Ddngthytngtrncdngnhduhiunutrn.Taschna=3,b=1.Khitngtrnsbng(31)16=216Vd11:(HHngHi2000)Chngminhrng:C+C+C++02n32322n4...3C42n=2(2+)12n2n2n2n12nGii:(1+x)=C2n+C2nx+C2nx2+...+C2n1x2n1+C2nx2n(1)2n0122n2n(1x)=C2nC2nx+C2nx2+...C2n1x2n1+C2nx2n(2)2n0122n2nLy(1)+(2)tac:(1+x)+(1x)2n2n=22n+C2nx2++C2nx2nC02...2n(4)+(2)2n2n=2C2n+C22n32+...+C2n32n02n24n+22n=C2n+C2n32+...+C22nn32n022Chnx=3suyra:22n(22n+1)=C2n+C2n32+...+C2nn32n02222n12(22n+1)=C2n+C2n32+...+C2n32n022nPCM2.Sdngohmcp1,2.a.ohmcp1.Duhiu:Khihsngtrcthptngdnhocgimdnt1,2,3,,nhayn,,3,2,1tclshngcdngkChockCabthtacthdngohmcpknknnkk11tnh.Cth:(a+x)n=Cnan+2Cnanx++nCnaxn011...nLyohmhaivtheoxtac:n(a+x)=Cnan+2Cnan2++nCnaxn(1)n1112n1...nythayx,abnghngsthchhptactngcntm.Vd12:(HBKHN1999)TnhtngCn2Cn+Cn4Cn++()12n1334...1nnCn8.Tinhiutiliu,thihntibookbooming.comGii:TathytngcntnhcdngnhVP(1).Viccnlichcnchna=1,x=1tatnhctngbng0.Cchkhc:SdngngthckC=nCtatnhctngbng:knkn11nCn+nCn++)...(1nCn1=n(1)n1nn1nCn01112111=0Vd13:Tnhtng:2008C2007+02007C2007++20071...C2007Gii:Hstrcthpgimdnt2008,2007,,1nndngohmliudhiu:(x+)20071=C2007x2007+2007x2006++C20070C1...2007Byginuolyohmthchc2007Cxtrongkhin2008dotaphi020072006nhnthmvixvongthctrnrimidngohm:x(x+1)2007=C2007x2008+C2007x2007+...+C2007x012007(x+1)(2008x+1)=2008C2007x2007+2007C2007x2006+...+C20072006012007Thayx=1votatmctngl2009.22006b.ohmcp2.Duhiu:Khihsngtrcthpcdng1.2,2.3,,(n1)nhay(n1)n,,3.2,2.1hay12,22,,n2(khngkdu)tccdngk(k1)Cahaytngquthnknnkk(k)Cnankbk1kthtacthdngohmncp2tnh.Xtathc(a+bx)n=Cn+Cnanbx++Cnbnxn011...nKhiohmhaivtheoxtac:bn(a+)n1bx=Cnanb+2Cnanb2x...+nCnbnxn1122n1ohmlnna:bn(n)(a+21bxn2)=2.1C2nanb2++(n)Cnbnxn(2)2...n1n1nytagnnhgiiquytxongvdtonchvicthaya,b,xbicchngsthchhpnathi.Vd14:(HANCSKhiA1998)Cho(x)=(1+x),(2n)nfa.Tnhf(1)b.Chngminhrng:2.1Cn+23.2Cn++n)nCn=n(n)2n3...(1n12Gii:a.f(x)=n(1+x)(x)=n(n)(1+x)n1n2f1(1)=n(1+x)nf2b.Tac9.Tinhiutiliu,thihntibookbooming.comnnf(x)=(1+x)=Cnxk=Cn+Cnx+Cnxkkn01kk=1k=2nf(x)=Cn+kCnxk11kk=2nf(x)=k(k1)Cnxk2kk=2nf(1)=k(k1)Cn=2n2kk=12.1C+3.2Cn+...+(p+1)Cnp+...+(n+1)nCn=n(n1)22n1(21nn+PCM)Tcubthay(n1)=(n+1)thtacmtbitonkhc:b.Chngminhrng:2.1C+C++(n+)pC++(n+)nC=n(n+)23.2...11n...12n1npnnn2Vibitonnytagiinhsau:Xtnhthc:(1+x)=C+Cx+...+Cxn01nnnnnNhn2vcangthcvingthilyohmcp2haivtheobinxtac:x02n(1+x)+(n)x(1+x)3.2Cnx++n+)nCnxn...(n1n2n1=2Cnx+121n1Chox=2tacPCMBitppdngBi1:(CSPBnTreKhiA2002)Chngminhrng:C20+20++20=2191C1...C1932004+1Bi2:(CKhiTM2004)Chngminhrng:C2004+22C2004+...+22004C2004=0120042Bi3:(HKTQD2000)Chng



minh:(2+x)...nCn=n.3n(n)n=1.2n.Cn+112.2n.Cn+223.2n.Cn++22n11Bi4:Rtgntng:12C200922008+2C200922007++122...20092C20092009III.Mtsphngphpkhc:0mknVd15:(HQGTP.HCM1997)Chok,m,nZChngminh:Cn.Cm+nCm++nCm=n+k0Ck11...CkmmCkmGii:(1+x)m=Cm+Cmx+...+Cmxm01mTac:(1+x)=Cnxn+Cnxn1+...+Cnn01n(1+x)m+n=Cm+n+Cm+nx+...+Cm+nxm+n01m+nSuyrahsxktrong(1+x)n.(1+x)mlCmCn+mCn++mCn0kC1k1...CmkmVhsxktrongkhai(1+x)m+nlkCm+n10.Tinhiutiliu,thihntibookbooming.comngnhtthc:(1+x)n.(1+x)m=(1+x)n+mTac:C.C+C++C...CC=knC0mkn11mknmmmkn+mPCMVd16:(2TH&TT2008)S2=(C)+2(Cn)+...+n(Cn)2221n2nvinlstnhinlGii:Tac:22()n1n21n+1n21+(Cn)+(n1)(Cnn1)Cn+n(Cn)22n2S=1+...+Cn+22(n(Cn)+(Cn)+...+(Cn1)122n2)+n2(=n(Cn+1)+(Cn)+...+(C))+nn222n12n2Sn=n(Cn)+(Cn)+...+(Cn)+n1222n2Mtkhctac:(1+x)hscaxnl:2n=C2n+C2nx++C2nx2n01...2nnC2n(*)Trongkhi:(1+x)n=Cn+Cnx+...+Cnxn01n(C)+(Cn)+...+(Cn)222Nnhscaxnl1n2n(**)T(*)v(**)C2n1=n(Cn)+(Cn)+...+(Cn)n1222n2nnSn=CPCM2n2BitppdngBi1:Chngminhrng:a)C3+2C3++nC1nn1...2nn1nn=.4nn1(HLut2001)b)1C+2C+...+nC=n(n+)221n122n2nnn2(1TH&TT2008)Bi2:Tnhcctngsau:a)C+C+C++29.2C3.21305.2...2330453028293012nCnCnnCnb)Cn0+...+(1)23n+13nBi3:tTk=().ChngminhTk=0k+113kC6n+2k1k=1

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