试题分析:(Ⅰ) 直接代入即可求值(Ⅱ)用后一项减前一项,结果和0作比较。结果等于0,说明是常数列;结果大于0,说明是递增数列;结果小于0说明是递减数列。注意讨论。(Ⅱ)先求数列数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032940918712.png)
的通项公式,再用作差法判断数列的增减性,再求其最值。
试题解析:(Ⅰ)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032940933533.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032940949556.png)
. .2分
(Ⅱ)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032941105920.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032941105833.png)
.
则当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032941136466.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032941136570.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032940964490.png)
时,数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032940824481.png)
为递增数列,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032940871505.png)
;
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032940996438.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032941245553.png)
,数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032940824481.png)
为递减数列,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032940871505.png)
. .7分
(Ⅲ)由上问可得,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240329412761108.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032940871505.png)
.
令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032941308615.png)
,即求数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032941308450.png)
的最大项和最小项.
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240329413391105.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032941339164.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032941354766.png)
.
则数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032941308450.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032941136466.png)
时递减,此时
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032941401610.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032941417591.png)
;
数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032941308450.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032940996438.png)
时递减,此时
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032941464629.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032941495663.png)
.
因此数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032941308450.png)
的最大项为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032941526500.png)
,最小项为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032941542491.png)
. . .13分