题目
题型:不详难度:来源:
已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210118-33093.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210118-25819.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210119-19076.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210119-89444.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210119-72156.png)
(1)当
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210119-73535.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210118-25819.png)
(2)在(1)的条件下,证明对
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210120-98244.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210120-42740.png)
(3)若
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210120-47338.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210120-98244.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210120-43063.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210120-32736.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210121-56609.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210119-72156.png)
(2)可以用裂项法求和进而证明也可以用数学归纳法证明
(3)可以用基本不等式证明也可以用导数证明,还可以利用数列的单调性证明
解析
试题分析:(1)当
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210119-73535.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210121-76729.png)
两边取倒数,得
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210121-37712.png)
故数列
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210122-91353.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210122-64019.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210122-85647.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210121-56609.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210119-72156.png)
(2)证法1:由(1)知
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210121-56609.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210122-67159.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210123-22232.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210123-76236.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210123-24106.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210123-15652.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210123-30432.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210124-50912.png)
[证法2:①当n=1时,等式左边
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210124-44112.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210124-55103.png)
②假设当
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210124-47906.png)
即
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210125-98245.png)
则当
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210125-24539.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210125-17305.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210125-84102.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210126-47633.png)
这就是说当
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210125-24539.png)
综①②知对于
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210120-98244.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210126-33125.png)
(3)当
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210120-47338.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210126-93178.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210126-37078.png)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210120-43063.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210127-95282.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210127-66713.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210127-35322.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210127-38022.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210128-14052.png)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210128-68386.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210128-60759.png)
即对
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210120-98244.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210120-32736.png)
【证法二:当
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210120-47338.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210126-93178.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210129-60661.png)
又
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210129-45115.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210129-88471.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210129-52014.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210130-99134.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210130-74119.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210130-19174.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210130-21367.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210130-50355.png)
【证法三:当
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210120-47338.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210126-93178.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210131-33281.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210131-27019.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210131-70788.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210132-74193.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210132-61132.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210132-25892.png)
所以命题得证 ……14分】
点评:本小题比较综合,既考查了数列的通项公式的求解,也考查了数列的前n项的求解,还考查了数列的性质的应用以及基本不等式、导数等的综合应用,难度较大,要求学生具有较高的分析问题、转化问题、解决问题的能力和运算求解能力.
核心考点
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210112-51737.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210113-13798.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210113-30544.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210055-46079.png)
A.P(k)对k=2013成立 | B.P(k)对每一个自然数k成立 |
C.P(k)对每一个正偶数k成立 | D.P(k)对某些偶数可能不成立 |
①
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210047-68320.png)
②
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210048-89484.png)
③
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210048-15948.png)
归纳出一般的结论,并用数学归纳法证明之.
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206210014-32190.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206205952-20717.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206205952-79611.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200206/20200206205952-31614.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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