1、数列综合测试题经典含标准答案数列综合测试题第I卷(选择题共60分)、选择题(本大题共12个小题,每小题5分,共60分,在每小题给出的四个选项中,只有一项是符号题目要求的。 )1已知等差数列an的前S3 S2n项和为Sn,且满足3 2 1,则数列 an的公差是( )1A.qB. 1C. 2D. 32.设等比数列an的前n项和为Sn,若8a2 + a5= 0,则下列式子中数值不能确定的是( )a5A a3S5B.S3an +1 C.anSn+ 1 D S3.(理)已知数列an满足* 1log3an+ 1 = log3an + 1(n N )且 a2+ a4 + a6 = 9,贝U Iog?(a5+
2、 a7 +a9)的值是( )A. 5B -1C. 5An 7n + 45 an4.已知两个等差数列an和bn的前n项和分别为An和Bn,且a = ,则使得.为Bn n + 3 bn正偶数时,n的值可以是()A. 1B. 2C. 5D . 3 或 115.已知a0, b0 , A为a, b的等差中项,正数 G为a, b的等比中项,贝U ab与AG的大小关系是( )A. ab= AGB. ab AGC. abwAGD.不能确定1a3 + a46.各项都是正数的等比数列 an的公比qz 1,且a2, oa3, a1成等差数列,则 的2a4+ a5值为( )1- *5 A.-B.号C躬-1C. 2D.
3、亍或号A. 24 B. 25C. 26 D. 278.数列an是等差数列,公差dM 0,且a2046 + ai978 a2012 = 0, bn是等比数列,且b2012 =a2012,贝y b2010 b2014 =( )A . 0 B. 1C. 4 D. 89. 已知各项均为正数的等比数列 an的首项ai = 3,前三项的和为 21,则a3 + a4 + a5=( )A. 33 B. 72C. 84 D. 18910.已知等差数列an的前n项和为Sn,若a1 = 1, S3= a5, am= 2011,贝U m =( )A. 1004 B. 1005C. 1006 D. 100711.设an
4、是由正数组成的等差数列, bn是由正数组成的等比数列,且 a1= b1, a2003=b2003,则( )A . a1002b1002 B . a1002 = b1002C . a1002 b1002 D . a1002 2)anM 0,两边同除 an+1an20 .解:(1 ) 2an+1 2an+an+1an=01 1 an 21 1数列二是首项为公差为-的等差数列(2)(n 1)da n ai1 n / KIXan i= ,(n N)n 11 n-bn=f(an 1)=f( )= n+6 (n N)n 1(3) n+6 (n 6, n N)n(b1 6 n) n (11 n)(n 6,
5、n N)n 11n 60221.解析(1)当 n = 1 时,au S1= 2,当 n 2 时,an= Sn Sn 1 = n(n + 1) (n 1)n = 2n,知 a1 = 2 满足该式 .数列an的通项公式为an= 2n.b1 b2 b3 bn (2)bn + 1an = 3 + 32+ 1+ 33+ 1 + + 3n+ 1(n1)bnb1 b2 b3an+1 = + c + c + + + 3+ 1 32+ 1 33+ 1 3n+1 3n+1+ 1bn + 1一得, 匸 =an+1 an= 2, bn+1= 2(3n+1+ 1),3n+1+ 1故 bn= 2(3n+ 1)(n N*)
6、.anbn n n(3)Cn = 丁 = n(3n+ 1)= n 3n + n,Tn= C1+ C2+ C3+ + cn= (1 x 3 + 2x 3 + 3x 33+ + nX 3n) + (1 + 2+ + n)令 Hn= 1 X 3 + 2 X 32 + 3 X 33+ - + nx 3n,则 3Hn= 1 X 32 + 2 X 33 + 3 X 34+ + nX 3n+1 “3 1 3n “一得,一2Hn= 3 + 32+ 33+ 3n nx 3n+ 1= nx 3n+11 32n 1 x 3n+ 1 + 3 H n =数列Cn的前n项和Tn =2n- 1 x 3n+ 1 + 3 n n + 1+22 解.an2n (n 2,且 n N )an2n1(n 2,且 n N数列an是等差数列,公差为d1,首项貝(2)由(1)得an (n(3) Snan2n丄)21-2(n1)d(n 1) 1 n212,2n222312Sn 22224(n1(n -) 221(1)2n1(1) (2)得Sn 1 2223(n1)2n22232n (n 2)2n1 12(1 2n)1 2(n2* 1(32n)2n3.Sn (2n 3) 2n(23)2n,2nbn= 131 4(3)b2, b4, b6b2n是首项为3,公比3 1 2 3的等比数列,
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