天津市2018高考数学（文）二轮复习：题型练4大题专项 数列的通项、求和问题

题型练4　大题专项(二) 数列的通项、求和问题 1.设数列{an}的前n项和为Sn,满足(1-q)Sn+qan=1,且q(q-1)≠0. (1)求{an}的通项公式; (2)若S3,S9,S6成等差数列,求证:a2,a8,a5成等差数列. 2.已知等差数列{an}的首项a1=1,公差d=1,前n项和为Sn,bn=. (1)求数列{bn}的通项公式; (2)设数列{bn}前n项和为Tn,求Tn. 3.(2017江苏,19)对于给定的正整数k,若数列{an}满足:an-k+an-k+1+…+an-1+an+1+…+an+k-1+an+k=2kan对任意正整数n(n>k)总成立,则称数列{an}是“P(k)数列”. (1)证明:等差数列{an}是“P(3)数列”; (2)若数列{an}既是“P(2)数列”,又是“P(3)数列”,证明:{an}是等差数列. 4.已知等差数列{an}的前n项和为Sn,公比为q的等比数列{bn}的首项是,且a1+2q=3,a2+4b2=6,S5=40. (1)求数列{an},{bn}的通项公式an,bn; (2)求数列的前n项和Tn. 5.已知函数f(x)=,数列{an}满足:2an+1-2an+an+1an=0,且anan+1≠0.在数列{bn}中,b1=f(0),且bn=f(an-1). (1)求证:数列是等差数列; (2)求数列{|bn|}的前n项和Tn. 6.记U={1,2,…,100}.对数列{an}(n∈N*)和U的子集T,若T=⌀,定义ST=0;若T={t1,t2,…,tk},定义ST=+…+.例如:T={1,3,66}时,ST=a1+a3+a66.现设{an}(n∈N*)是公比为3的等比数列,且当T={2,4}时,ST=30. (1)求数列{an}的通项公式; (2)对任意正整数k(1≤k≤100),若T⊆{1,2,…,k},求证:ST<ak+1; (3)设C⊆U,D⊆U,SC≥SD,求证:SC+SC∩D≥2SD. ## 题型练4　大题专项(二) 数列的通项、求和问题 1.解 (1)当n=1时,由(1-q)S1+qa1=1,a1=1. 当n≥2时,由(1-q)Sn+qan=1,得(1-q)Sn-1+qan-1=1,两式相减,得an=qan-1. 又q(q-1)≠0,所以{an}是以1为首项,q为公比的等比数列,故an=qn-1