试题分析:(1)关于
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023216931396.png)
和
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217133353.png)
的递推式,一般有两种方法可解决,1:转化为项的递推式,根据递推式 直接求通项公式,2:转化为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023216931396.png)
的递推关系,先求
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023216931396.png)
,再求通项公式,该题已知数列前n项和
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023216931396.png)
和
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217133353.png)
的递推关系,由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217196581.png)
可的
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217133353.png)
与
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217227392.png)
的关系,然后由等差数列定义证明,知道等差数列后再求通项公式;
(2)①将
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217243507.png)
代入不等式,解不等式可得,②恒成立问题往往可以采取参变分离的方法,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217258564.png)
或
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217274567.png)
的形式,最后转化为求函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217289464.png)
最值,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217305685.png)
或
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217321715.png)
,该题可转化为求
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217336378.png)
的最大值问题,求
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217336378.png)
的最大值可以结合函数的函数或者单调性处理,但是注意定义域
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217367525.png)
.
试题解析:(1)令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217383357.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217399588.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217414513.png)
,由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240232174301354.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240232174451596.png)
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217414513.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217477973.png)
,
即数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023216884494.png)
是以2为首项,2为公差的等差数列, ∴
(2)①
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240232175231448.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217555808.png)
②∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217570952.png)
,又∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217087445.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023216993532.png)
∴各项中数值最大为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217601393.png)
,∵对一切正整数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023216915304.png)
,总有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217024564.png)
恒成立,因此
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824023217102542.png)
.