(本小题主要考查等差数列、等比数列和不等式等基础知识,考查运算求解能力、推理论证能力,以及函数与方程、化归与转化等数学思想.)
(1)解法1:当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258824244.gif)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258887840.gif)
,…………………………………………2分
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258902503.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258918414.gif)
.……………………………………………………………………………………4分
所以数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258965489.gif)
是首项为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258980283.gif)
的常数列.……………………………………………………………5分
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258996402.gif)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175259012368.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175259043466.gif)
.
所以数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258590381.gif)
的通项公式为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175259012368.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175259043466.gif)
.………………………………………………………7分
解法2:当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258824244.gif)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258887840.gif)
,…………………………………………2分
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175259136516.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258918414.gif)
.…………………………………………………………………………………4分
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231752593861654.gif)
.………………………5分
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258637256.gif)
,符合
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175259417220.gif)
的表达式.…………………………………………………………………………6分
所以数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258590381.gif)
的通项公式为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175259012368.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175259043466.gif)
.………………………………………………………7分
(2)假设存在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258668199.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175259807619.gif)
,使得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258731217.gif)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258762346.gif)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258778344.gif)
成等比数列,
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175259885412.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175259948353.gif)
.…………………………………………………………………………………………8分
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175259963554.gif)
(
n≥2),
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231752599941770.gif)
………………………………11分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231753000261072.gif)
.…………………………………13分
这与
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175259885412.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175259948353.gif)
矛盾.
故不存在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258668199.gif)
(
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258684511.gif)
),使得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258731217.gif)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258762346.gif)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175258778344.gif)
成等比数列.…………………………………14分