题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041038-19562.gif)
①
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041039-34878.gif)
②存在实数M,使
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041039-38502.gif)
(I)在只有5项的有限数列
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041039-48595.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041039-77537.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041039-16749.gif)
(II)设
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041040-25071.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041040-66284.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041040-81833.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041040-18363.gif)
(III)设数列
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041040-27250.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041041-86400.gif)
求证:数列
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041041-98208.gif)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041038-19562.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041041-12173.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041040-18363.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041041-38479.gif)
解析
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041038-19562.gif)
取
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041042-32995.gif)
故
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041038-19562.gif)
对于数列
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041041-12173.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041042-97392.gif)
不仅有
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041042-99226.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041042-55251.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041042-89190.gif)
显然满足集合W的条件①②,
故
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041041-12173.gif)
(II)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041043-33389.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041040-66284.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041043-80547.gif)
设其公比为q>0,
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041043-27460.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041043-50039.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041043-85195.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041044-17885.gif)
对于
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041044-63977.gif)
且
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041044-86892.gif)
故
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041040-18363.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041041-38479.gif)
(III)证明:(反证)若数列
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041044-94170.gif)
使
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041045-49673.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041045-17910.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041045-49673.gif)
假设
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041045-13732.gif)
当n=m+1时,由
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041045-74552.gif)
而
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041045-46412.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041046-52328.gif)
所以,对于任意的
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041046-52090.gif)
显然
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041046-53161.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041046-67400.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041046-18374.gif)
所以假设不成立,故命题得证. …………14分
核心考点
试题【(14分)设集合W由满足下列两个条件的数列构成:①②存在实数M,使(n为正整数)(I)在只有5项的有限数列;试判断数列是否为集合W的元素;(II)设是各项为正的】;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/20191013/20191013041034-27852.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041034-81185.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041035-67659.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041035-94103.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041035-88240.png)
A.24 | B.42 | C.60 | D.78 |
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041011-99877.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041011-25176.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041011-98052.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041012-85982.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041012-77361.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041006-99417.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041007-48677.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041007-46988.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041007-35495.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041007-46988.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013041007-53799.png)
A.10,8 | B.13,29 | C.13,8 | D.10,29 |
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013040955-80752.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013040955-83463.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013040956-50064.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013040956-72995.png)
A.![]() | B.8 | C.![]() | D.4 |
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013040951-46265.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013040951-93682.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013040951-11035.png)
A.等差数列 | B.等比数列 |
C.既是等差数列又是等比数列 | D.既非等差数列又非等比数列 |
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