2024年数学高考大一轮复习第六章 §6.6　数列中的综合问题

§6.6　数列中的综合问题

考试要求　数列的综合运算问题以及数列与函数、不等式等知识的交汇问题，是历年高考的热点内容．一般围绕等差数列、等比数列的知识命题，涉及数列的函数性质、通项公式、前n项和公式等．

题型一　等差数列、等比数列的综合运算

例1 (2023·厦门模拟)已知数列{an}的前n项和为Sn，且Sn＝n2＋n，递增的等比数列{bn}满足b1＋b4＝18，b2·b3＝32.

(1)求数列{an}，{bn}的通项公式；

(2)若cn＝an·bn，n∈N*，求数列{cn}的前n项和Tn.

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思维升华 数列的综合问题常将等差、等比数列结合，两者相互联系、相互转化，解答这类问题的方法：寻找通项公式，利用性质进行转化．

跟踪训练1 (2022·全国甲卷)记Sn为数列{an}的前n项和．已知＋n＝2an＋1.

(1)证明：{an}是等差数列；

(2)若a4，a7，a9成等比数列，求Sn的最小值．

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题型二　数列与其他知识的交汇问题

命题点1　数列与不等式的交汇

例2 (1)已知数列{an}满足a1＋a2＋a3＋…＋an＝n2＋n(n∈N*)，设数列{bn}满足：bn＝，数列{bn}的前n项和为Tn，若Tn<λ(n∈N*)恒成立，则实数λ的取值范围为(　　)

A.   B.

C.   D.

听课记录：______________________________________________________________

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(2)已知数列{an}满足a1＝，3an，2an＋1，anan＋1成等差数列．

①证明：数列是等比数列，并求{an}的通项公式；

②记{an}的前n项和为Sn，求证：≤Sn<.

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命题点2　数列与函数的交汇

例3 (1)(2023·龙岩模拟)已知函数f(x)＝x3＋4x，记等差数列{an}的前n项和为Sn，若f(a1＋2)＝100，f(a2 022＋2)＝－100，则S2 022等于(　　)

A．－4 044   B．－2 022

C．2 022   D．4 044

(2)数列{an}是等差数列，a1＝1，公差d∈[1，2]，且a4＋λa10 ＋a16＝15，则实数λ的最大值为________．

听课记录：______________________________________________________________

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思维升华 (1)数列与不等式的综合问题及求解策略

①判断数列问题的一些不等关系，可以利用数列的单调性比较大小或借助数列对应的函数的单调性比较大小．

②以数列为载体，考查不等式恒成立的问题，此类问题可转化为函数的最值．

③考查与数列有关的不等式证明问题，此类问题一般采用放缩法进行证明，有时也可通过构造函数进行证明．

(2)数列与函数交汇问题的主要类型及求解策略

①已知函数条件，解决数列问题，此类问题一般利用函数的性质、图象研究数列问题．

②已知数列条件，解决函数问题，解决此类问题一般要利用数列的通项公式、前n项和公式、求和方法等对式子化简变形．

跟踪训练2 (1)设{an}是等比数列，函数y＝x2－x－2 023的两个零点是a2，a3，则a1a4等于(　　)

A．2 023  B．1  C．－1  D．－2 023

(2)数列{an}满足a1＝1，an＋1＝2an(n∈N*)，Sn为其前n项和．数列{bn}为等差数列，且满足b1＝a1，b4＝S3.

①求数列{an}，{bn}的通项公式；

②设cn＝，数列{cn}的前n项和为Tn，证明：≤Tn<.

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