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1、(全國通用版)2022高考數(shù)學(xué)二輪復(fù)習(xí) 中檔大題規(guī)范練(二)數(shù)列 理1(2018三明質(zhì)檢)已知正項數(shù)列an的前n項和為Sn,a11,且(t1)Sna3an2(tR)(1)求數(shù)列an的通項公式;(2)若數(shù)列bn滿足b11,bn1bnan1,求數(shù)列的前n項和Tn.解(1)因為a11,且(t1)Sna3an2,所以(t1)S1a3a12,所以t5.所以6Sna3an2.當(dāng)n2時,有6Sn1a3an12,得6ana3ana3an1,所以(anan1)(anan13)0,因為an0,所以anan13,又因為a11,所以an是首項a11,公差d3的等差數(shù)列,所以an3n2(nN*)(2)因為bn1bnan
2、1,b11,所以bnbn1an(n2,nN*),所以當(dāng)n2時,bn(bnbn1)(bn1bn2)(b2b1)b1anan1a2b1.又b11也適合上式,所以bn(nN*)所以,所以Tn,.2(2018葫蘆島模擬)設(shè)等差數(shù)列an的前n項和為Sn,且S3,S4成等差數(shù)列,a53a22a12.(1)求數(shù)列an的通項公式;(2)設(shè)bn2n1,求數(shù)列的前n項和Tn.解(1)設(shè)等差數(shù)列an的首項為a1,公差為d,由S3,S4成等差數(shù)列,可知S3S4S5,得2a1d0,由a53a22a12,得4a1d20,由,解得a11,d2,因此,an2n1(nN*)(2)令cn(2n1)n1,則Tnc1c2cn,Tn1
3、1352(2n1)n1,Tn13253(2n1)n,得Tn12(2n1)n12 (2n1)n 3,Tn6(nN*)3(2018廈門質(zhì)檢)已知等差數(shù)列an滿足(n1)an2n2nk,kR.(1)求數(shù)列an的通項公式;(2)設(shè)bn,求數(shù)列bn的前n項和Sn.解(1)方法一由(n1)an2n2nk,令n1,2,3,得到a1,a2,a3,an是等差數(shù)列,2a2a1a3,即,解得k1.由于(n1)an2n2n1(2n1)(n1),又n10,an2n1(nN*)方法二an是等差數(shù)列,設(shè)公差為d,則ana1d(n1)dn(a1d),(n1)an(n1)(dna1d)dn2a1na1d,dn2a1na1d2n
4、2nk對于nN*均成立,則解得k1,an2n1(nN*)(2)由bn111,得Snb1b2b3bn1111nnn(nN*)4(2018天津河?xùn)|區(qū)模擬)已知等比數(shù)列an滿足條件a2a43(a1a3),a2n3a,nN*.(1)求數(shù)列an的通項公式;(2)數(shù)列bn滿足n2,nN*,求bn的前n項和Tn.解(1)設(shè)an的通項公式為ana1qn1(nN*),由已知a2a43(a1a3),得a1qa1q33(a1a1q2),所以q3.又由已知a2n3a,得a1q2n13aq2n2,所以q3a1,所以a11,所以an的通項公式為an3n1(nN*)(2)當(dāng)n1時,1,b11,當(dāng)n2時,n2,所以(n1)2
5、,由得2n1,所以bn(2n1)3n1,b11也符合,綜上,bn(2n1)3n1(nN*)所以Tn130331(2n3)3n2(2n1)3n1,3Tn131332(2n3)3n1(2n1)3n,由得2Tn1302(31323n1)(2n1)3n13023(2n1)3n13n3(2n1)3n(22n)3n2,所以Tn1(n1)3n(nN*)5(2018宿州模擬)已知數(shù)列an的前n項和為Sn,數(shù)列Sn的前n項和為Tn,滿足Tn2Snn2.(1)證明數(shù)列an2是等比數(shù)列,并求出數(shù)列an的通項公式;(2)設(shè)bnnan,求數(shù)列bn的前n項和Kn.解(1)由Tn2Snn2,得a1S1T12S11,解得a1S11,由S1S22S24,解得a24.當(dāng)n2時,SnTnTn1 2Snn22Sn1(n1)2,即Sn2Sn12n1,Sn12Sn2n1,由得an12an2,an122(an2),又a222(a12),數(shù)列an2是以a123為首項,2為公比的等比數(shù)列,an232n1,即an32n12(nN*)(2)bn3n2n12n,Kn3(120221n2n1)2(12n)3(120221n2n1)n2n.記Rn120221n2n1,2Rn121222(n1)2n1n2n,由,得Rn2021222n1n2nn2n (1n)2n1,Rn(n1)2n1.Kn3(n1)2nn2n3(nN*)