题目内容
已知二次函数
,其导函数为
,数列
的前
项和为
点
均在函数
的图像上;.
(Ⅰ)求数列
的通项公式;
(Ⅱ)若
,求数列
的通项公式;
(Ⅲ)已知不等式
成立,
求证:![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232324081741697.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232407908915.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232407940678.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232407955456.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232407986297.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232408018416.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232408049846.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232408064562.png)
(Ⅰ)求数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232407955456.png)
(Ⅱ)若
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232324081111536.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232408127471.png)
(Ⅲ)已知不等式
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232408142757.png)
求证:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232324081741697.png)
解:(Ⅰ)数列
的通项公式:![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232408205604.png)
(Ⅱ)数列
的通项公式:![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232324082521322.png)
(Ⅲ)略
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232407955456.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232408205604.png)
(Ⅱ)数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232408127471.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232324082521322.png)
(Ⅲ)略
本试题主要是考查了函数与数列以及不等式的综合运用。
(1)因为二次函数
,则
,得到参数a,b的值。
(2)由(Ⅰ)得,
,
,然后利用前n项和与通项公式的关系式得到结论。
(3)已知不等式
成立,故
则
……1分
所以
,运用放缩法证明不等式
(1)因为二次函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232407908915.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232408298884.png)
(2)由(Ⅰ)得,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232408314910.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232324083301073.png)
(3)已知不等式
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232408142757.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232408376482.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232408423662.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232324084391131.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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