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1、2022年高考數(shù)學(xué)二輪復(fù)習(xí) 專題突破練13 求數(shù)列的通項(xiàng)及前n項(xiàng)和 理
1.(2018河南鄭州一模,理17)已知等差數(shù)列{an}的前n項(xiàng)和為Sn,且a2+a5=25,S5=55.
(1)求數(shù)列{an}的通項(xiàng)公式;
(2)設(shè)anbn=,求數(shù)列{bn}的前n項(xiàng)和Tn.
2.已知{an}為公差不為零的等差數(shù)列,其中a1,a2,a5成等比數(shù)列,a3+a4=12.
(1)求數(shù)列{an}的通項(xiàng)公式;
(2)記bn=,設(shè){bn}的前n項(xiàng)和為Sn,求最小的正整數(shù)n,使得Sn>.
3.(2018山西太原三模,17)已知數(shù)列{an}滿足a1
2、=,an+1=.
(1)證明數(shù)列是等差數(shù)列,并求{an}的通項(xiàng)公式;
(2)若數(shù)列{bn}滿足bn=,求數(shù)列{bn}的前n項(xiàng)和Sn.
4.(2018江西上饒三模,理17)已知等比數(shù)列{an}的前n項(xiàng)和為Sn,且6Sn=3n+1+a(n∈N*).
(1)求a的值及數(shù)列{an}的通項(xiàng)公式;
(2)若bn=(3n+1)an,求數(shù)列{an}的前n項(xiàng)和Tn.
5.已知數(shù)列{an}滿足a1=1,a2=3,an+2=3an+1-2an(n∈N*).
(1)證明:數(shù)列{an+1-an}是等比數(shù)列;
(2)求數(shù)列{an}的通項(xiàng)公
3、式和前n項(xiàng)和Sn.
6.已知等差數(shù)列{an}滿足:an+1>an,a1=1,該數(shù)列的前三項(xiàng)分別加上1,1,3后成等比數(shù)列,an+2log2bn=-1.
(1)求數(shù)列{an},{bn}的通項(xiàng)公式;
(2)求數(shù)列{an·bn}的前n項(xiàng)和Tn.
7.(2018寧夏銀川一中一模,理17)設(shè)Sn為數(shù)列{an}的前n項(xiàng)和,已知an>0,+2an=4Sn+3.
(1)求{an}的通項(xiàng)公式:
(2)設(shè)bn=,求數(shù)列{bn}的前n項(xiàng)和.
8.設(shè)Sn是數(shù)列{an}的前n項(xiàng)和,an>0,且4Sn
4、=an(an+2).
(1)求數(shù)列{an}的通項(xiàng)公式;
(2)設(shè)bn=,Tn=b1+b2+…+bn,求證:Tn<.
參考答案
專題突破練13 求數(shù)列的通項(xiàng)及前n項(xiàng)和
1.解 (1)
求得an=3n+2.
(2)bn=
Tn=b1+b2+…+bn=+…+,∴Tn=
2.解 (1)設(shè)等差數(shù)列{an}的公差為d,
∵a1,a2,a5成等比數(shù)列,a3+a4=12,
即
∵d≠0,∴解得
∴an=2n-1,n∈N*.
(2)∵bn=,∴Sn=1-+…+=1-
令1-,解得n>1 008,
故所求的n=1 009.
3.(1)證明 ∵an+
5、1=,
=2,是等差數(shù)列,+(n-1)×2=2+2n-2=2n,即an=
(2)解 ∵bn=,∴Sn=b1+b2+…+bn=1++…+,
則Sn=+…+,
兩式相減得Sn=1++…+=2,
∴Sn=4-
4.解 (1)∵6Sn=3n+1+a(n∈N*),
∴當(dāng)n=1時,6S1=6a1=9+a;
當(dāng)n≥2時,6an=6(Sn-Sn-1)=2×3n,即an=3n-1.∵{an}為等比數(shù)列,
∴a1=1,則9+a=6,a=-3,
∴{an}的通項(xiàng)公式為an=3n-1.
(2)由(1)得bn=(3n+1)3n-1,
Tn=b1+b2+…+bn=4×30+7×31+…+(3n+1
6、)3n-1,①
3Tn=4×31+7×32+…+(3n-2)3n-1+(3n+1)3n,②
由①-②,得-2Tn=4+32+33+…+3n-(3n+1)3n,
-2Tn=4+-(3n+1)3n,
-2Tn=,
∴Tn=
5.(1)證明 ∵an+2=3an+1-2an(n∈N*),
∴an+2-an+1=2(an+1-an)(n∈N*),=2.
∵a1=1,a2=3,∴數(shù)列{an+1-an}是以a2-a1=2為首項(xiàng),公比為2的等比數(shù)列.
(2)解 由(1)得,an+1-an=2n(n∈N*),
∴an=(an-an-1)+(an-1-an-2)+…+(a2-a1)+a1=2n
7、-1+2n-2+…+2+1=2n-1,(n∈N*).
Sn=(2-1)+(22-1)+(23-1)+…+(2n-1)=(2+22+23+…+2n)-n=-n=2n+1-2-n.
6.解 (1)設(shè)等差數(shù)列{an}的公差為d,且d>0,由a1=1,a2=1+d,a3=1+2d,分別加上1,1,3后成等比數(shù)列,得(2+d)2=2(4+2d),解得d=2,∴an=1+(n-1)×2=2n-1.∵an+2log2bn=-1,
∴l(xiāng)og2bn=-n,即bn=
(2)由(1)得an·bn=Tn=+…+,①
Tn=+…+,②
①-②,得Tn=+2+…+
∴Tn=1+=3-=3-
7.解 (1)
8、由+2an=4Sn+3,可知+2an+1=4Sn+1+3.
兩式相減,得+2(an+1-an)=4an+1,即2(an+1+an)==(an+1+an)(an+1-an).
∵an>0,∴an+1-an=2.
+2a1=4a1+3,
∴a1=-1(舍)或a1=3.
則{an}是首項(xiàng)為3,公差d=2的等差數(shù)列,∴{an}的通項(xiàng)公式an=3+2(n-1)=2n+1.
(2)∵an=2n+1,∴bn=,
∴數(shù)列{bn}的前n項(xiàng)和Tn=+…+
8.(1)解 4Sn=an(an+2),①
當(dāng)n=1時,4a1=+2a1,即a1=2.
當(dāng)n≥2時,4Sn-1=an-1(an-1+2).②
由①-②得4an=+2an-2an-1,即2(an+an-1)=(an+an-1)·(an-an-1).∵an>0,∴an-an-1=2,
∴an=2+2(n-1)=2n.
(2)證明 ∵bn=,
∴Tn=b1+b2+…+bn=1-+…+1-<