本题考查数列与不等式的综合,综合性强,难度较大.易错点是基础知识不牢固,不会运用数列知识进行等价转化转化.解题时要认真审题,注意挖掘题设中的隐含条件.
1)设二次函数f(x)=ax
2+bx.f'(x)=2ax+b,由2a=6b=-2,知f(x)=3x
2-2x,由(n,Sn)在y=3x
2-2x上,知S
n=3n
2-2n.由此能求出数列{a
n}的通项公式.
(2)由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224554470678.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224554486373.png)
是数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224554502471.png)
的前
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224554361277.png)
项和,求使得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224554533676.png)
,得到m>20T
n,解不等式得到结论。
解: (1) 由题意得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224554892711.png)
, 即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224554907671.png)
,………………1分
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224554923435.png)
时 ,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232245549381561.png)
,…………4分
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224554970357.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224554985702.png)
, ………………5分
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232245550011091.png)
, ……………………6分
(2) 由(1)得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232245550321762.png)
,…………………8分
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232245550631658.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224555079749.png)
. ……………………11分
因此,使得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232245550941254.png)
成立的
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224554564339.png)
必须且只需满足
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224555141632.png)
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224555157486.png)
,
故满足要求的的最小正整数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224554595453.png)
………………13分