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单元质检六数列(B)(时间:45分钟满分:100分)一、选择题(本大题共6小题,每小题7分,共42分)1.已知等差数列an的公差和首项都不等于0,且a2,a4,a8成等比数列,则a1+a5+a9a2+a3=()A.2B.3C.5D.7答案B解析设an的公差为d.由题意,得a42=a2a8,(a1+3d)2=(a1+d)(a1+7d),d2=a1d.d0,d=a1,a1+a5+a9a2+a3=15a15a1=3.2.在单调递减的等比数列an中,若a3=1,a2+a4=52,则a1=()A.2B.4C.2D.22答案B解析设an的公比为q.由已知,得a1q2=1,a1q+a1q3=52,q+q3q2=52,q2-52q+1=0,q=12(q=2舍去),a1=4.3.(2018河北唐山期末)在数列an中,a1=1,an+1=2an,Sn为an的前n项和.若Sn+为等比数列,则=()A.-1B.1C.-2D.2答案B解析由题意,得an是等比数列,公比为2,Sn=2n-1,Sn+=2n-1+.Sn+为等比数列,-1+=0,=1,故选B.4.(2018陕西西安八校联考)设等差数列an的前n项和为Sn,若S6S7S5,则满足SnSn+1S7S5,6a1+652d7a1+762d5a1+542d,a70,S13=13(a1+a13)2=13a70,满足SnSn+10,两边取以2为底的对数可得log2(an+1)=log2(an-1+1)2=2log2(an-1+1),则数列log2(an+1)是以1为首项,2为公比的等比数列,log2(an+1)=2n-1,an=22n-1-1,又an=an-12+2an-1(n2),可得an+1=an2+2an(nN*),两边取倒数可得1an+1=1an2+2an=1an(an+2)=121an-1an+2,即2an+1=1an-1an+2,因此bn=1an+1+1an+2=1an-1an+1,所以Sn=b1+bn=1a1-1an+1=1-122n-1,故答案为1-122n-1.三、解答题(本大题共3小题,共44分)9.(14分)已知数列an的前n项和为Sn,首项为a1,且12,an,Sn成等差数列.(1)求数列an的通项公式;(2)数列bn满足bn=(log2a2n+1)(log2a2n+3),求数列1bn的前n项和Tn.解(1)12,an,Sn成等差数列,2an=Sn+12.当n=1时,2a1=S1+12,即a1=12;当n2时,an=Sn-Sn-1=2an-2an-1,即anan-1=2,故数列an是首项为12,公比为2的等比数列,即an=2n-2.(2)bn=(log2a2n+1)(log2a2n+3)=(log222n+1-2)(log222n+3-2)=(2n-1)(2n+1),1bn=12n-112n+1=1212n-1-12n+1.Tn=121-13+13-15+12n-1-12n+1=121-12n+1=n2n+1.10.(15分)已知数列an和bn满足a1=2,b1=1,2an+1=an,b1+12b2+13b3+1nbn=bn+1-1.(1)求an与bn;(2)记数列anbn的前n项和为Tn,求Tn.解(1)2an+1=an,an是公比为12的等比数列.又a1=2,an=212n-1=12n-2.b1+12b2+13b3+1nbn=bn+1-1,当n=1时,b1=b2-1,故b2=2.当n2时,b1+12b2+13b3+1n-1bn-1=bn-1,-,得1nbn=bn+1-bn,得bn+1n+1=bnn,故bn=n.(2)由(1)知anbn=n12n-2=n2n-2.故Tn=12-1+220+n2n-2,则12Tn=120+221+n2n-1.以上两式相减,得12Tn=12-1+120+12n-2-n2n-1=21-12n1-12-n2n-1,故Tn=8-n+22n-2.11.(15分)设an是等比数列,公比大于0,其前n项和为Sn(nN*),bn是等差数列.已知a1=1,a3=a2+2,a4=b3+b5,a5=b4+2b6.(1)求an和bn的通项公式;(2)设数列Sn的前n项和为Tn(nN*),求Tn;证明k=1n(Tk+bk+2)bk(k+1)(k+2)=2n+2n+2-2(nN*).(1)解设等比数列an的公比为q.由a1=1,a3=a2+2,可得q2-q-2=0.因为q0,可得q=2,故an=2n-1.设等差数列bn的公差为d.由a4=b3+b5,可得b1+3d=4.由a5=b4+2b6,可得3b1+13d=16,从而b1=1,d=1,故bn=n.所以,数列an的通项公式为an=2n-1,数列bn的通项公式为bn=n.(2)解由(1),有Sn=1-2n1-2=2n-1,故Tn=k=1n(2k-1)=k=1n2k-n=2(1-2n)1-2-n=2n+1-n-2.证明因为(Tk+bk+2)bk(k+1)(k+2)=(2k+1-k-2+k+2)k(k+1)(k+2)=k2k+1(k+1)(k+2)=2k+2k+2-2k+1k+1,所以,k=1n(Tk+bk+2)bk(k+1)(k+2)=233-222+244-233+2n+2n+2-2n+1n+1=2n+2n+2-2.
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