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1、专题突破练(三)A级基础达标练一、选择题1(2015贵阳模拟)设Sn为等差数列an的前n项和,若a33,S9S627,则该数列的首项a1等于()A B C. D.解析由得解得a1.答案D2在等差数列an中,a1a2a33,a18a19a2087,则此数列前20项的和等于()A290 B300 C580 D600解析由a1a2a33a23得a21,由a18a19a203a1987,得a1929,所以S2010(a2a19)300.答案B3已知ann,把数列an的各项排列成如图31的三角形状,a1a2a3a4a5a6a7a8a9图31记A(m,n)表示第m行的第n个数,则A(10,12)()A.9
2、3 B.92 C.94 D.112解析前9行共有1351781项,所以A(10,12)为数列中的第811293项,所以a9393.答案A4已知等差数列an的公差d2,a1a4a7a9750,那么a3a6a9a99的值是()A78 B82 C148 D182解析a3a6a9a99(a12d)(a42d)(a72d)(a972d)a1a4a7a972d335066(2)82.答案B5已知Sn为数列an的前n项和,且满足2ana1S1Sn(a10,nN*),则a7等于()A16 B32 C64 D128解析令n1,则a11,当n2时,2a21S21a2,解得a22,当n2时,由2an1Sn,得2an
3、11Sn1,两式相减,解得2an2an1an,即an2an1,于是数列an是首项为1,公比为2的等比数列,因此an2n1.故a72664.答案C二、填空题6设lg an成等差数列,公差dlg 3,且lg an的前三项和为6lg 3,则an的通项公式为_解析由题意知lg a1lg a2lg a33lg a26lg 3,lg a22lg 3,又公差dlg 3,lg a1lg 3,lg anlg 3(n1)lg 3nlg 3lg 3n,an3n.答案an3n7(2013课标全国卷)若数列an的前n项和Snan,则an的通项公式是an_.解析当n1时,S1a1,a11.当n2时,anSnSn1an(a
4、nan1),an2an1,即2,an是以1为首项的等比数列,其公比为2,an1(2)n1,即an(2)n1.答案(2)n18数列an的前n项和Snn24n2,则|a1|a2|a10|_.解析当n1时,a1S11.当n2时,anSnSn12n5.an令2n50,得n,当n2时,an0,|a1|a2|a10|(a1a2)(a3a4a10)S102S266.答案66三、解答题9(2015东北三省四市模拟)设数列an的前n项和为Sn2n1,数列bn满足bnn,(1)求数列an的通项公式;(2)求数列bn的前n项和Tn.解(1)当n1时,a1S14.由Sn2n1得Sn12n(n2),anSnSn12n1
5、2n2n(n2),an(2)当n1时,b11,T1.当n2时,bnnnn,Tn(234n)(1234n),上式对于n1也成立,Tn.10已知单调递增的等比数列an满足:a2a3a428,且a32是a2,a4的等差中项(1)求数列an的通项公式;(2)若bnanlogan,Snb1b2bn,求使Snn2n150成立的正整数n的最小值解(1)设等比数列an的首项为a1,公比为q,依题意,有2(a32)a2a4,代入a2a3a428,得a38,a2a420,解之得或又an单调递增,q2,a12,an2n.(2)bn2nlog2nn2n,Sn12222323n2n,2Sn122223324(n1)2n
6、n2n1,得Sn222232nn2n1n2n12n1n2n12,Snn2n150,即2n1250,2n152,又当n4时,2n1253252.故使Snn2n150成立的正整数n的最小值为5.B级能力提升练1设等差数列an的前n项和为Sn,且满足S150,S160,得a80.由S160,得a9a80,所以a90,所以dS14S10.又a1a2a80a9a10a15,所以最大的项为,故选D.答案D2已知数列an的前n项和为Sn,且Sn2nan,则数列an的通项公式an_.解析由于Sn2nan,所以Sn12(n1)an1,后式减去前式,得Sn1Sn2an1an,即an1an1,变形为an12(an2),则数列an2是以a12为首项,为公比的等比数列又a12a1,即a11.则an2(1)n1,所以an2n1.答案2n13已知数列an的前n项和为Sn,且满足a1,an2SnSn1(n2)(1)求数列an的通项公式an;(2)求证:SSS.解(1)an2SnSn1(n2),SnSn12SnSn1.两边同除以SnSn1,得2(n2),数列是以2为首项,以d2为公差的等差数列,(n1)d22(n1)2n,Sn.将Sn代入an2SnSn1,得an(2)证明S(n2),S,当n2时,SSS;当n1时,S.综上,SSS.