题目内容
已知数列
的前n项和为
,且满足
,
.
(Ⅰ)问:数列
是否为等差数列?并证明你的结论;
(Ⅱ)求
和
;
(Ⅲ)求证:
.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212829703457.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212829718388.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212829734473.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212829734951.png)
(Ⅰ)问:数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212829749553.png)
(Ⅱ)求
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212829718388.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212829859348.png)
(Ⅲ)求证:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232128298741147.png)
(1)见解析;(2)
,
;(3)见解析.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212829890624.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232128299211475.png)
本题主要考查递推数列、等差数列与不等式的综合应用,考查分类讨论思想,考查放缩的方法
解:(1)由已知有
,
;
时,![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212830030749.png)
所以
,即
是以2为首项,公差为2 的等差数列.
(2)由(1)得:
,![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212829890624.png)
当
时,![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212830155648.png)
.
当
时,
,所以![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232128299211475.png)
(3)当
时,
,成立.
当
时,![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232128304361795.png)
=![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212830451962.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232128304831130.png)
综上有
.
解:(1)由已知有
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212829952614.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212829968509.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212830015437.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212830030749.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212830046699.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212829749553.png)
(2)由(1)得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212830108927.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212829890624.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212830015437.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212830155648.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212830171665.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212830264357.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212830280485.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232128299211475.png)
(3)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212830264357.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212830405769.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212830015437.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232128304361795.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212830451962.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232128304831130.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212830498868.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232128298741147.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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