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1、專題三數(shù)列、推理與證明第1講等差數(shù)列、等比數(shù)列(推薦時(shí)間:50分鐘)一、選擇題1已知數(shù)列an是公比為q的等比數(shù)列,且a1,a3,a2成等差數(shù)列,則公比q的值為()A1或 B1C D22已知等比數(shù)列an中,a4a610,則a1a72a3a7a3a9的值等于()A10 B20 C60 D1003(2011大綱全國(guó))設(shè)Sn為等差數(shù)列an的前n項(xiàng)和,若a11,公差d2,Sk2Sk24,則k等于()A8 B7 C6 D54等比數(shù)列an的前n項(xiàng)和為Sn,已知S1,2S2,3S3成等差數(shù)列,則an的公比為()A2 B3 C. D.5設(shè)an是公差為正數(shù)的等差數(shù)列,若a1a2a315,a1a2a380,則a11
2、a12a13的值為()A120 B105C90 D756(2011遼寧)若等比數(shù)列an滿足anan116n,則公比為()A2 B4 C8 D16二、填空題7(2011湖南)設(shè)Sn是等差數(shù)列an (nN*)的前n項(xiàng)和,且a11,a47,則S5_.8已知數(shù)列an的前n項(xiàng)和Sn滿足log2(Sn1)n1,則數(shù)列an的通項(xiàng)公式是_9(2010江蘇)函數(shù)yx2(x0)的圖象在點(diǎn)(ak,a)處的切線與x軸的交點(diǎn)的橫坐標(biāo)為ak1,其中kN*,a116,則a1a3a5的值是_10設(shè)等差數(shù)列an的各項(xiàng)均為整數(shù),其公差d0,a56,若a3,a5,am (m5)是公比為q (q0)的等比數(shù)列,則m的值為_三、解答題
3、11已知數(shù)列an的首項(xiàng)a1a,anan11(nN*,n2)若bnan2(nN*)(1)問數(shù)列bn是否能構(gòu)成等比數(shù)列?并說明理由(2)若已知a11,設(shè)數(shù)列anbn的前n項(xiàng)和為Sn,求Sn.12.(2011大綱全國(guó))設(shè)等比數(shù)列an的前n項(xiàng)和為Sn,已知a26,6a1a330,求an和Sn.13已知數(shù)列an滿足an2an12n1(n2),且a481.(1)求數(shù)列的前三項(xiàng)a1,a2,a3;(2)求證:數(shù)列為等差數(shù)列,并求an.答案1A 2D3D4D5B6B725 8an 921 101111解(1)b1a2,anbn2,所以bn2(bn12)1,bnbn1.所以,當(dāng)a2時(shí),數(shù)列bn能構(gòu)成等比數(shù)列;當(dāng)a2時(shí),數(shù)列bn不能構(gòu)成等比數(shù)列(2)當(dāng)a1,得bn()n1,an2()n1,anbn()n12()n1,所以Sn2(1)4(1).12解設(shè)an的公比為q,由題設(shè)得解得或當(dāng)a13,q2時(shí),an32n1,Sn3(2n1);當(dāng)a12,q3時(shí),an23n1,Sn3n1.13(1)解an2an12n1,a42a3241.又a481,a333,同理:a213,a15.(2)證明由an2an12n1(n2),得1,1,是等差數(shù)列;的公差d1.(n1)1n1.an(n1)2n1.