试题分析:(1)利用基本量思想求解两个数列的通项公式,然后才有错位相减法求解数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652683547.png)
的前
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652699276.png)
项和;(2)利用
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652746463.png)
等量关系关系,减少公差d,进而将
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652777332.png)
与
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652792348.png)
进行表示,然后才有作差比较进行分析,注意分类讨论思想的应用.
试题解析:(1)依题意,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653026865.png)
,
故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653042898.png)
,
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653058803.png)
, 3分
令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240206530731165.png)
, ①
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240206530891464.png)
, ②
①
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653104164.png)
②得,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240206531201331.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240206531361144.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653151759.png)
,
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652839966.png)
. 7分
(2)因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652746463.png)
,
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653198708.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653214679.png)
,
故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653229908.png)
,
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653260496.png)
, 9分
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240206532761234.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240206532921205.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240206533072030.png)
11分
(ⅰ)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652855470.png)
时,由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653338356.png)
知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240206533702353.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240206534161276.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653432997.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653448301.png)
, 13分
(ⅱ)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652886420.png)
时,由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653338356.png)
知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240206534942374.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240206535101308.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653526784.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653541303.png)
,
综上所述,当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652855470.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652870477.png)
;当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652886420.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652902475.png)
;当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652917456.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652933456.png)
. 16分
(注:仅给出“
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652855470.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652870477.png)
;
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652886420.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652902475.png)
”得2分.)
方法二:(注意到数列的函数特征,运用函数性质求解)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653900841.png)
(易知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653916389.png)
),
令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653931706.png)
,有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653947671.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653962781.png)
,
令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653994823.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654072774.png)
.记
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654087818.png)
.
若
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654103427.png)
,则在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654134446.png)
上
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654150537.png)
,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654181429.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654134446.png)
上为单调增函数,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654212653.png)
,
这与
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653947671.png)
相矛盾;
若
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654259501.png)
,则在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654274435.png)
上
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654290547.png)
,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654181429.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654274435.png)
上为单调减函数,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654555654.png)
,
这与
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653947671.png)
相矛盾;
所以,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654586574.png)
.
故在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654602459.png)
上
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654290547.png)
,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654181429.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654274435.png)
上为单调减函数,
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654696519.png)
上
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654150537.png)
,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654181429.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654742494.png)
上为单调增函数.
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020653947671.png)
,所以,当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654774536.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654914529.png)
,当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654930457.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654961513.png)
,
所以,当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652886420.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020654992575.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020655023480.png)
,
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652855470.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020655054578.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020655070480.png)
,
综上所述,当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652855470.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652902475.png)
;当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652886420.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652870477.png)
;当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652917456.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020652933456.png)
.