试题分析:(1)首先设出公差和公比,根据已知条件及等比数列和等差数列的性质,列方程组解方程组,求得公差和公比,写出各自的通项公式;(2)因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057776297.png)
取偶数和奇数时,数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057761491.png)
的项数会有变化,所以对
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057776297.png)
分取偶数和奇数两种情况进行讨论,根据等差数列和等比数列的前
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057776297.png)
项和公式,求出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057917373.png)
的表达式,根据
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057917373.png)
前后两项的变化确定
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057917373.png)
的单调性,求得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057917373.png)
每种情况下的最小值,比较一下,取两个最小值中的较小者.
试题解析:(1)设数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023058213389.png)
的公差是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023058229321.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057761491.png)
的公比为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023058260310.png)
,
由已知得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240230582751084.png)
,解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023058291758.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057948615.png)
; 2分
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240230583381342.png)
,解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023058353408.png)
或
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023058353502.png)
(舍去),所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057963492.png)
; .4分
(2)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057776297.png)
为偶数时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240230584001184.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240230584311009.png)
,
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057776297.png)
为奇数时
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240230584631301.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240230584781112.png)
. .10分
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057776297.png)
为偶数时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240230585092055.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057917373.png)
先减后增,
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023058650421.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023058650998.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023058681500.png)
;
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023058681430.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023058712959.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023058728507.png)
;
所以当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057776297.png)
为偶数时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057917373.png)
最小值是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023058775510.png)
. 12分
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057776297.png)
为奇数时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240230588062102.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057917373.png)
先减后增,
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023058837376.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023058853950.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023058868465.png)
,
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023058884429.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240230589931025.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023059009505.png)
,
所以当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057776297.png)
为奇数时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057917373.png)
最小值是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023058026534.png)
.
比较一下这两种情况下的
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057917373.png)
的最小值,可知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057917373.png)
的最小值是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023058026534.png)
. .14分
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023057776297.png)
项和公式;2、数列与函数单调性的综合应用;3、数列与求函数最值的综合运用;4、数列的函数特性.