本试题主要是考查了数列的通项公式的求解,以及数列的概念和数列的单调性的运用。
(1)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803896413.png)
时,有累加法得到
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232238039111142.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803958956.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803974370.png)
也满足上式,
所以数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803209457.png)
的通项为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803818539.png)
.
(2)因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803350833.png)
,
所以对任意的
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804067518.png)
有
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804083927.png)
,
所以数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804098466.png)
是一个以6为周期的循环数列
进而证明
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804130427.png)
为常数列
(3)因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803443823.png)
,且
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803365569.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804192881.png)
,
且对任意的
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804067518.png)
,有
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232238042231109.png)
,
设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804254735.png)
,(其中
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804270263.png)
为常数且
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804286794.png)
),所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232238043171766.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232238043321721.png)
,
所以数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804364546.png)
均为以7为公差的等差数列.记
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804395566.png)
,构造新数列来分析周期性和最值问题。
(1)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803896413.png)
时,有
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232238044261084.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803942721.png)
……………………1分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803958956.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803974370.png)
也满足上式,
所以数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803209457.png)
的通项为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803818539.png)
. ………………………………………………………3分
(2)因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803350833.png)
,
所以对任意的
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804067518.png)
有
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804613906.png)
,
所以数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804098466.png)
是一个以6为周期的循环数列……………………………………………………5分
又因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803365569.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232238047071285.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232238047221398.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232238047381484.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804769545.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804785494.png)
,
所以数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803412435.png)
为常数列. ……………………………………………………………………7分
(3)因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803443823.png)
,且
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803365569.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804192881.png)
,
且对任意的
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804067518.png)
,有
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232238042231109.png)
,
设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804254735.png)
,(其中
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804270263.png)
为常数且
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804286794.png)
),所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232238050191827.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232238050341755.png)
,
所以数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804364546.png)
均为以7为公差的等差数列.……………………………………………10分
记
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804395566.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232238051122510.png)
,
(其中
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223805128533.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223805144508.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223804270263.png)
为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223805190707.png)
中的一个常数),
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223805206586.png)
时,对任意的
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223805128533.png)
有
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223805253468.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223805268409.png)
;…………………………………………12分
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223805284616.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232238053151442.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232238053311220.png)
①若
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223805471613.png)
,则对任意的
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223805690451.png)
有
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223805705504.png)
,数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223805908793.png)
为单调减数列;
②若
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223805924603.png)
,则对任意的
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223805690451.png)
有
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223805970525.png)
,数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223805908793.png)
为单调增数列;
综上,当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232238060021086.png)
时,数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803724582.png)
中必有某数重复出现无数次……………14分
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223806048306.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223806080497.png)
符合要求;当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223806095352.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223806111713.png)
符合要求,此时的
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223806142647.png)
;
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223806158340.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223806189719.png)
符合要求,此时的
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223806204892.png)
;
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223806220327.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223806251723.png)
符合要求,此时的
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223806282788.png)
;
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223806298338.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223806314727.png)
符合要求,此时的
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223806329868.png)
;
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223806360360.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223806376641.png)
符合要求,此时的
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223806392889.png)
;
即当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803849940.png)
时,数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823223803724582.png)
中必有某数重复出现无数次.………………………16分