(本小题主要考查等差数列、等比数列、不等式等基础知识,考查方程思想以及运算求解能力.)
解:(1)设等差数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132369381.gif)
的公差为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132775198.gif)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132790689.gif)
.………………………………………1分
由已知,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231751328211110.gif)
………………………………………………………………………3分
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132837747.gif)
解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132868495.gif)
…………………………………………………………………………5分
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132884573.gif)
(
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132899386.gif)
).………………………………………………………………6分
(2)假设存在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132400204.gif)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132416199.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132962653.gif)
,使得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132650207.gif)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132681323.gif)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132697210.gif)
成等比数列,
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133040433.gif)
.……………………………………………………………………………………………7分
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133071595.gif)
,…………………………………………………………………………………8分
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133087800.gif)
.
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133102757.gif)
.……………………………………………………………………………9分
整理,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133133613.gif)
.…………………………………………………………………………10分
以下给出求
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132400204.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132416199.gif)
的三种方法:
方法1:因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133196258.gif)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133211523.gif)
.………………………………………………………11分
解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133227562.gif)
.……………………………………………………………………………12分
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133258499.gif)
,
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133274349.gif)
,此时
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133289250.gif)
.
故存在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133274349.gif)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133289250.gif)
,使得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132650207.gif)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132681323.gif)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132697210.gif)
成等比数列.……………………………………………14分
方法2:因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133414368.gif)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133445682.gif)
.…………………………………………………11分
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133461640.gif)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133477615.gif)
.
解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133492497.gif)
或
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133523480.gif)
.………………………………………………………………12分
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133258499.gif)
,
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133274349.gif)
,此时
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133289250.gif)
.
故存在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133274349.gif)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133289250.gif)
,使得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132650207.gif)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132681323.gif)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132697210.gif)
成等比数列.……………………………………………14分
方法3:因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133711415.gif)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133726672.gif)
.……………………………………………11分
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133742633.gif)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133757697.gif)
.
解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133773621.gif)
或
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133789626.gif)
.…………………………………………………12分
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133258499.gif)
,
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133274349.gif)
,此时
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133289250.gif)
.
故存在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133274349.gif)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175133289250.gif)
,使得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132650207.gif)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132681323.gif)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823175132697210.gif)
成等比数列.……………………………………………14分