题目内容
在数列
中,
,其中
.
(Ⅰ)求证:数列
为等差数列;
(Ⅱ)求证:![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232037237201521.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823203723611455.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232037236581038.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823203723673500.png)
(Ⅰ)求证:数列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823203723689466.png)
(Ⅱ)求证:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232037237201521.png)
Ⅰ)证明:
∴数列
为等差数列
(Ⅱ)因为
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823203723970761.png)
原不等式即为证明
,
即
成立
用数学归纳法证明如下:
当![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823203724032395.png)
时,
成立,所
以
时,原不等式成立
假设当
时,
成立
当
时,![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232037242041323.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232037242352356.png)
当
时,不等式成立,所以对
,总有
成立
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232037237361521.png)
∴数列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823203723689466.png)
(Ⅱ)因为
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823203723782650.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823203723814670.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823203723970761.png)
原不等式即为证明
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232037239851479.png)
即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232037240161462.png)
用数学归纳法证明如下:
当
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823203724032395.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823203724048164.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823203724079590.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823203724094164.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823203724032395.png)
假设当
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823203724141412.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823203724157950.png)
当
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823203724188452.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232037242041323.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232037242352356.png)
当
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823203724188452.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823203724282659.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823203724297972.png)
略
![](http://thumb.zyjl.cn/images/loading.gif)
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