题目
题型:不详难度:来源:
设数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233851-25382.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233851-87073.gif)
(1)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233852-12032.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233852-53291.gif)
(2)令
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233852-95183.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233853-69427.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233853-32531.gif)
答案
(1)证明略
(2)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233853-93919.gif)
解析
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233853-12965.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233853-14258.gif)
假设
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233854-22076.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233854-56736.gif)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233854-60532.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233854-51014.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233855-75843.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233855-14145.gif)
综上由数学归纳法可知,
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233852-12032.gif)
证法二:当
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233853-12965.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233855-71257.gif)
假设
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233854-22076.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233854-56736.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233854-60532.gif)
由函数
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233855-36327.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233855-43189.gif)
因此只需证:
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233856-53265.gif)
而这等价于
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233856-66357.gif)
显然成立,所以当
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233854-60532.gif)
综上由数学归纳法可知,
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233852-12032.gif)
证法三:由递推公式得
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233856-24990.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233857-84022.gif)
上述各式相加并化简得
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233857-40423.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233857-88929.gif)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233853-12965.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233852-12032.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233857-45228.gif)
(2)解法一:
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233857-63486.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233858-86507.gif)
又显然
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233858-98515.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233853-93919.gif)
解法二:
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233858-57942.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233858-89782.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233859-23897.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233859-45015.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233859-91815.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233859-67188.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233859-40199.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233900-61242.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233853-93919.gif)
解法三:
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233900-89720.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233900-34118.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233900-64207.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233901-93979.gif)
故
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233901-46194.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233853-93919.gif)
核心考点
举一反三
已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233808-45336.gif)
(1)若函数
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233809-40817.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233809-44734.gif)
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233809-68200.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233809-61348.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233810-65811.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233810-18681.gif)
(3)设各项为正的数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233810-55818.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233810-31983.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233810-84130.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233754-93670.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233754-99846.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233754-37952.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233755-89662.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233755-62032.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233745-36047.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233745-88854.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233746-73835.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233746-28923.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233746-42385.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233746-80511.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233739-20468.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233739-34650.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233739-11998.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233739-16666.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012233740-42009.png)
A.0 | B.![]() | C.![]() | D.2 |
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