2020版高考數(shù)學一輪復習 課時規(guī)范練31 數(shù)列求和 理 北師大版.doc

課時規(guī)范練31數(shù)列求和基礎(chǔ)鞏固組1.數(shù)列1,3,5,7116,(2n-1)+12n,的前n項和Sn的值等于()A.n2+1-12nB.2n2-n+1-12nC.n2+1-12n-1D.n2-n+1-12n2.(2018河北衡水中學金卷十模,3)已知數(shù)列an是各項為正數(shù)的等比數(shù)列,點M(2,log2a2),N(5,log2a5)都在直線y=x-1上,則數(shù)列an的前n項和為()A.2n-2B.2n+1-2C.2n-1D.2n+1-13.(2018山東濰坊二模,4)設數(shù)列an的前n項和為Sn,若Sn=-n2-n,則數(shù)列2(n+1)an的前40項的和為()A.3940B.-3940C.4041D.-40414.已知函數(shù)f(x)=xa的圖像過點(4,2),令an=1f(n+1)+f(n),nN+.記數(shù)列an的前n項和為Sn,則S2 018=.5.(2018浙江余姚中學4月模擬,17)已知等差數(shù)列an的前n項和為Sn,且S5=30,S10=110.(1)求Sn;(2)記Tn=1S1+1S2+1Sn,求Tn.6.(2018山西晉城月考)已知數(shù)列an滿足a1=3,an+1=2an+(-1)n(3n+1).(1)求證:數(shù)列an+(-1)nn是等比數(shù)列;(2)求數(shù)列an的前10項和S10.7.(2018山東濰坊一模,17)公差不為0的等差數(shù)列an的前n項和為Sn,已知S4=10,且a1,a3,a9成等比數(shù)列.(1)求an的通項公式;(2)求數(shù)列an3n的前n項和Tn.綜合提升組8.(2018廣東中山期末)等比數(shù)列an中,已知對任意自然數(shù)n,a1+a2+a3+an=2n-1,則a12+a22+a32+an2等于()A.2n-1B. (3n-1)C. (4n-1)D.以上都不對9.(2018湖北重點中學五模)設等差數(shù)列an的前n項和為Sn,a4=4,S5=15,若數(shù)列1anan+1的前m項和為1011,則m=()A.8B.9C.10D.1110.(2018山東濰坊三模,17)已知數(shù)列an的前n項和為Sn,且1,an,Sn成等差數(shù)列.(1)求數(shù)列an的通項公式;(2)若數(shù)列bn滿足anbn=1+2nan,求數(shù)列bn的前n項和Tn.11.(2018江西上饒三模,17)已知等比數(shù)列an的前n項和為Sn,且6Sn=3n+1+a(nN+).(1)求a的值及數(shù)列an的通項公式;(2)若bn=(3n+1)an,求數(shù)列an的前n項和Tn.創(chuàng)新應用組12.幾位大學生響應國家的創(chuàng)業(yè)號召,開發(fā)了一款應用軟件.為激發(fā)大家學習數(shù)學的興趣,他們推出了“解數(shù)學題獲取軟件激活碼”的活動.這款軟件的激活碼為下面數(shù)學問題的答案已知數(shù)列1,1,2,1,2,4,1,2,4,8,1,2,4,8,16,其中第一項是20,接下來的兩項是20,21,再接下來的三項是20,21,22,依此類推.求滿足如下條件的最小整數(shù)N:N>100且該數(shù)列的前N項和為2的整數(shù)冪.那么該款軟件的激活碼是()A.440B.330C.220D.11013.(2018云南玉溪月考)數(shù)列an滿足:a1=,a2=,且a1a2+a2a3+anan+1=na1an+1對任何的正整數(shù)n都成立,則1a1+1a2+1a97的值為()A.5 032B.5 044C.5 048D.5 050參考答案課時規(guī)范練31數(shù)列求和1.A該數(shù)列的通項公式為an=(2n-1)+12n,則Sn=1+3+5+(2n-1)+12+122+12n=n2+1-12n.2.C由題意log2a2=2-1=1,可得a2=2,log2a5=5-1=4,可得a5=16,a5a2=q3=8q=2,a1=1Sn=1-2n1-2=2n-1,故選C.3.DSn=-n2-n,a1=S1=-2.當n2時,an=Sn-Sn-1=-n2-n+(n-1)2+(n-1)=-2n,則數(shù)列an的通項公式為an=-2n,2(n+1)an=2-2n(n+1)=-1n-1n+1,數(shù)列2(n+1)an的前40項的和為S40=-1-12+12-13+140-141=-4041.4.2 019-1由f(4)=2,可得4a=2,解得a=,則f(x)=x12.an=1f(n+1)+f(n)=1n+1+n=n+1-n,S2 018=a1+a2+a3+a2 018=(2-1)+(3-2)+(4-3)+(2 019-2 018)=2 019-1.5.解 (1)設an的首項為a1,公差為d,由題意得S5=5a1+10d=30,S10=10a1+45d=110,解得a1=2,d=2,所以Sn=n2+n.(2)1Sn=1n(n+1)=1n-1n+1,所以Tn=1-12+12-13+1n-1n+1=1-1n+1=nn+1.6.(1)證明 an+1=2an+(-1)n(3n+1),an+1+(-1)n+1(n+1)an+(-1)nn=2an+(-1)n(3n+1)-(-1)n(n+1)an+(-1)nn=2an+(-1)nnan+(-1)nn=2.又a1-1=3-1=2,數(shù)列an+(-1)nn是首項為2,公比為2的等比數(shù)列.(2)解 由(1)得an+(-1)nn=22n-1=2n,an=2n-(-1)nn,S10=(2+22+210)+(1-2)+(3-4)+(9-10)=2(1-210)1-2-5=211-7=2 041.7.解 (1)設an的公差為d,由題設可得,4a1+6d=10,a32=a1a9,4a1+6d=10,(a1+2d)2=a1(a1+8d),解得a1=1,d=1.an=n.(2)令cn=n3n,則Tn=c1+c2+cn=13+232+333+n-13n-1+n3n,13Tn=132+233+n-13n+n3n+1,-得:23Tn=13+132+13n-n3n+1=13(1-13n)1-13-n3n+1=12-123n-n3n+1,Tn=34-2n+343n.8.C當n=1時,a1=21-1=1,當n2時,a1+a2+a3+an=2n-1,a1+a2+a3+an-1=2n-1-1,兩式做差可得an=2n-2n-1=2n-1,且n=1時,21-1=20=1=a1,an=2n-1,故an2=4n-1,a12+a22+a32+an2=1(1-4n)1-4=13(4n-1).9.CSn為等差數(shù)列an的前n項和,設公差為d,則a4=4,S5=15=5a3,解得d=1,則an=4+(n-4)1=n.由于1anan+1=1n(n+1)=1n-1n+1,則Sm=1-12+12-13+1m-1m+1=1-1m+1=1011,解得m=10.10.解 (1)由已知1,an,Sn成等差數(shù)列,得2an=1+Sn,當n=1時,2a1=1+S1=1+a1,a1=1.當n2時,2an-1=1+Sn-1,由-,得2an-2an-1=an,anan-1=2,數(shù)列an是以1為首項,2為公比的等比數(shù)列,an=a1qn-1=12n-1=2n-1.(2)由anbn=1+2nan得bn=1an+2n,Tn=b1+b2+bn=1a1+2+1a2+4+1an+2n=1a1+1a2+1an+(2+4+2n)=1-12n1-12+(2+2n)n2=n2+n+2-12n-1.11.解 (1)6Sn=3n+1+a(nN+),當n=1時,6S1=6a1=9+a;當n2時,6an=6(Sn-Sn-1)=23n,即an=3n-1,an為等比數(shù)列,a1=1,則9+a=6,a=-3,an的通項公式為an=3n-1.(2)由(1)得bn=(3n+1)3n-1,Tn=b1+b2+bn=430+731+(3n+1)3n-1,3Tn=431+732+(3n-2)3n-1+(3n+1)3n,-2Tn=4+32+33+3n-(3n+1)3n,Tn=(6n-1)3n+14.12.A設數(shù)列的首項為第1組,接下來兩項為第2組,再接下來三項為第3組,以此類推,設第n組的項數(shù)為n,則前n組的項數(shù)和為n(1+n)2.第n組的和為1-2n1-2=2n-1,前n組總共的和為2(1-2n)1-2-n=2n+1-2-n.由題意,N>100,令n(1+n)2>100,得n14且nN+,即N出現(xiàn)在第13組之后.若要使最小整數(shù)N滿足:N>100且前N項和為2的整數(shù)冪,則SN-Sn(1+n)2應與-2-n互為相反數(shù),即2k-1=2+n(kN+,n14),所以k=log2(n+3),解得n=29,k=5.所以N=29(1+29)2+5=440,故選A.13.Ba1a2+a2a3+anan+1=na1an+1,a1a2+a2a3+anan+1+an+1an+2=(n+1)a1an+2,-,得-an+1an+2=na1an+1-(n+1)a1an+2,n+1an+1-nan+2=4,同理得nan-n-1an+1=4,n+1an+1-nan+2=nan-n-1an+1,整理得2an+1=1an+1an+2,1an是等差數(shù)列.a1=14,a2=15,等差數(shù)列1an的首項為4,公差為1,1an=4+(n-1)1=n+3,1a1+1a2+1a97=97(4+100)2=5 044.