试题分析:(1)将
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807619680.png)
利用
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240238073231013.png)
进行化简,得到关于
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807650392.png)
与
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807682348.png)
的递推关系式,根据其特点,求其通项公式;(2)本题关键是求出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807697373.png)
,根据其表达式的特点,可每两项组合后提取公因式
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807713399.png)
后,转化为等差数列求和,但要注意对
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807728297.png)
,分奇偶性讨论,求出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807697373.png)
后,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807401588.png)
对
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807401523.png)
恒成立再分离参数后转化为求最值问题,容易求出实数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807416267.png)
的取值范围;(3)此类问题,一般先假设存在符合条件的数列,解出来则存在,如果得到矛盾的结果,则假设错误,这样的数列则不存在.
试题解析:⑴因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240238078222546.png)
,
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807838655.png)
. 2分
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807853371.png)
,所以数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807338481.png)
是以1为首项,公差为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807900382.png)
的等差数列.
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807572689.png)
. 4分
⑵①当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807931739.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240238079471615.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240238079781444.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807994940.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240238080091449.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023808040876.png)
. 6分
②当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023808056771.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240238080721010.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240238080871406.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240238081181476.png)
. 8分
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240238081342402.png)
要使
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807401588.png)
对
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807401523.png)
恒成立,
只要使
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240238081651016.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807728297.png)
为偶数恒成立.
只要使
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240238082121004.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807728297.png)
为偶数恒成立,故实数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807416267.png)
的取值范围为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807604612.png)
. 10分
⑶由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807572689.png)
,知数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807338481.png)
中每一项都不可能是偶数.
①如存在以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807432315.png)
为首项,公比
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023808337310.png)
为2或4的数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807494509.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807510518.png)
,
此时
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807494509.png)
中每一项除第一项外都是偶数,故不存在以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807432315.png)
为首项,公比为偶数的数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807494509.png)
. 12分
②当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023808446363.png)
时,显然不存在这样的数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807494509.png)
.
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023808493403.png)
时,若存在以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807432315.png)
为首项,公比为3的数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807494509.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807510518.png)
.
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023808618424.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023808633396.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023808649909.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023808664653.png)
.
所以满足条件的数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023807557497.png)
的通项公式为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023808664653.png)
. 16分