试题分析:(Ⅰ)设函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207603573.png)
的图像的顶点的纵坐标构成数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207618456.png)
,求证:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207618456.png)
为等差数列,由于
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207650562.png)
是二次函数,只需对
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207650562.png)
配方,确定函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207650562.png)
的图象的顶点的纵坐标,从而可求数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207821480.png)
的通项公式,由数列的通项公式,再证明数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207821480.png)
为等差数列;(Ⅱ))函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207650562.png)
的图像的顶点到
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207665275.png)
轴的距离构成数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207884479.png)
,求
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207884479.png)
的前
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207712277.png)
项和
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207712388.png)
,先确定数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207884479.png)
的通项公式
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207977775.png)
,显然数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207884479.png)
是等差数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207821480.png)
的每一项加上绝对值,像这一类题的解法,关键是找出变号项,进而可分段求出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207884479.png)
的前n项和
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207712388.png)
.
试题解析:(Ⅰ)∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240252080401651.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025208055608.png)
, 2分
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240252080861050.png)
,
∴数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207618456.png)
为等差数列. 4分
(Ⅱ)由题意知,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025207977775.png)
, 6分
∴当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025208133493.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025208133612.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240252081492090.png)
8分
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025208180422.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824025208196609.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240252082111420.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240252082271399.png)
. 10分
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240252077431703.png)
. 12分