题目
题型:不详难度:来源:
在单调递增数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062619-23607.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062619-27186.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062629-33665.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062629-39501.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062629-10846.png)
(Ⅰ)求
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062629-17041.png)
(Ⅱ)判断数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062619-23607.png)
(Ⅲ)设
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062630-42681.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062630-24880.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062629-10846.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062630-15891.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062630-11755.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062619-23607.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062630-42326.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062619-23607.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062631-37904.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062629-10846.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062629-33665.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062629-39501.png)
(3)通过前几项归纳猜想,然后运用数学归纳法加以证明。
解析
试题分析:(Ⅰ)解:因为
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062619-23607.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062631-40118.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062631-75979.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062632-44391.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062632-51804.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062632-12152.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062632-35870.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062630-11755.png)
(Ⅱ)证明:数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062619-23607.png)
用反证法证明:
假设数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062619-23607.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062630-42326.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062633-66010.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062633-35425.png)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062619-23607.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062631-37904.png)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062629-10846.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062629-33665.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062629-39501.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062629-10846.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062633-67651.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062633-56093.png)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062631-37904.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062633-76577.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062634-26325.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062634-69165.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062634-32842.png)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062634-66586.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062634-67126.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062633-76577.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062634-26325.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062634-69165.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062635-44318.png)
(Ⅲ)证明:观察:
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062635-33588.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062635-21998.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062635-57926.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062636-50774.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062636-45663.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062636-87427.png)
用数学归纳法证明:
(1)当
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062636-51621.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062636-52705.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062637-30108.png)
(2)假设当
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062637-66502.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062637-58596.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062637-31863.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062638-98829.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062638-86500.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062638-61982.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062638-74808.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062638-30892.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062639-13786.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062639-76961.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062639-56762.png)
根据(1)(2)可知,对任意
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062629-10846.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062636-87427.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062639-69797.png)
由已知得,
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062640-12386.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062640-24413.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062640-25890.png)
所以当
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062640-78964.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062640-40738.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062641-85055.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062641-11806.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062641-97325.png)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062641-52182.png)
所以对任意
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062642-46125.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062634-26325.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062642-30913.png)
对任意
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062642-46125.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062634-26325.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062642-64180.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062634-26325.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062642-94052.png)
因为数列{
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062642-71688.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062643-99843.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062643-81479.png)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062639-69797.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062630-15891.png)
点评:解决数列的单调性问题,要根据定义法来说明,同时要对于正面证明比较难的试题,要正难则反,属于中档题。
核心考点
试题【(本小题共14分)在单调递增数列中,,不等式对任意都成立.(Ⅰ)求的取值范围;(Ⅱ)判断数列能否为等比数列?说明理由;(Ⅲ)设,,求证:对任意的,.】;主要考察你对等差数列等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062611-35813.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062611-64098.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062611-10865.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062611-37998.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062612-73807.png)
A.15 | B.18 | C.9 | D.12 |
等差数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062604-65562.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062604-96883.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062604-97622.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062605-60579.png)
(Ⅰ)求
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062604-65562.png)
(Ⅱ)设
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062605-64598.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062605-73903.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062604-96883.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062605-24170.png)
已知数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062552-49510.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062552-65624.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062552-83269.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062552-30045.png)
(Ⅰ)求
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062553-30267.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062553-95663.png)
(Ⅱ)若
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062553-40696.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062553-10869.png)
(Ⅲ)若
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062553-42944.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062554-93495.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062554-13274.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062546-49845.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062546-25884.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062546-35325.png)
A.58 | B.88 | C.143 | D.176 |
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062522-87380.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062522-56004.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062522-86449.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062522-87380.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062523-39411.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062523-87131.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062523-76637.png)
(1)求数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062522-87380.png)
(2)令
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062523-31297.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062524-34209.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062523-39411.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012062524-97968.png)
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