试题分析:(Ⅰ)容易求得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012736595296.png)
. 1分
故可以猜想
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012736611352.png)
.下面利用数学归纳法加以证明:
显然当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012736751269.png)
时,结论成立. 2分
假设当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012736767215.png)
;
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012736783231.png)
时(也可以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012736798218.png)
),结论也成立,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012736814286.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012736814239.png)
. 3分
那么当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012736829237.png)
时,由题设与归纳假设可知:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240127368611369.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012736876304.png)
4分
即当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012736829237.png)
时,结论也成立,综上,对
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012736907249.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012736907280.png)
成立. 6分
(Ⅱ)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240127369231104.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012736939761.png)
, 8分
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240127369541306.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012736673321.png)
. 10分
所以只需要证明
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012736689428.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012736689351.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012737095674.png)
(显然成立)
所以对任意的自然数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012737110233.png)
,都有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012737126398.png)
. 12分
点评:数学归纳法用来证明与正整数有关的题目,证明步骤:1,证明当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012737141205.png)
时命题成立。2,假设当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012737157219.png)
时命题成立,借此证明当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824012737173235.png)
是命题成立,综上1,2得证;数列求和常用的方法有分组求和裂项相消求和错位相减求和等