試題分析:(1)要證數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030516839481.png)
是等比數(shù)列,可根據(jù)題設求出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517105424.png)
,當然也可再求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517105437.png)
,雖然得出的
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517120534.png)
成等比數(shù)列,但前面有限項成等比不能說明所有項都成等比,必須嚴格證明.一般方法是把已知式
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517136839.png)
中的
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030516839297.png)
用
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517183341.png)
代換得到
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517198915.png)
,兩式相減得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517214820.png)
,這個式子中把
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030516839297.png)
用
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517245336.png)
代換又得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517261793.png)
,兩式再相減,正好得出數(shù)列的前后項關系的遞推關系
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517276529.png)
,正是等比數(shù)列的表現(xiàn).(2)由題間
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517276478.png)
,對不等式
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030516949703.png)
用分離參數(shù)法得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517307712.png)
,求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030516995292.png)
的最小值就與求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517339562.png)
的最大值(也只要能是取值范圍)聯(lián)系起來了.(3)只能由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517027752.png)
成等差數(shù)列列出唯一的等式,這個等式是關于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517042403.png)
的二元方程,它屬于不定方程,有無數(shù)解,只是由于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517042403.png)
都是正整數(shù),利用正整數(shù)的性質(zhì)可得出具體的解.
試題解析:(1)當n=1時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517401370.png)
;當n=2時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517417508.png)
當n
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517432242.png)
3時,有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240305174321242.png)
得:
化簡得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517214820.png)
3分
又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517261793.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517276529.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030516839481.png)
是1為首項,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517510334.png)
為公比的等比數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517058870.png)
6分
(2)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240305175261575.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240305175571021.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517557411.png)
11分
(3)若三項成等差,則有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240305175731054.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517604444.png)
,右邊為大于2的奇數(shù),左邊為偶數(shù)或1,不成立
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030517073546.png)
16分