题目内容
(本小题满分13分)
已知数列![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220221528678.png)
满足
,且当![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220221684536.png)
时,
,令
.
(Ⅰ)写出
的所有可能的值;
(Ⅱ)求
的最大值;
(Ⅲ)是否存在数列
,使得
?若存在,求出数列
;若不存在,说明理由.
已知数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220221528678.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220221575732.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220221637506.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220221684536.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220221887574.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220221903646.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220221934903.png)
(Ⅰ)写出
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222090556.png)
(Ⅱ)求
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222105568.png)
(Ⅲ)是否存在数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222121366.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222152945.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222121366.png)
(1)
的所有可能的值为:
,
,
,
,
.(2)
的最大值为
;(3)
.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222090556.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222215248.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222246291.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222261262.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222277302.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222293257.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222105568.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222339612.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222355942.png)
第一问中,根据题意可知当i=5时,满足条件的数列
的所有可能情况有
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222449497.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222480456.png)
,分别结算得到![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222636641.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222870639.png)
的值
第二问中,因为递推关系可知由
,
可设
,则
或
(
,
),
那么借助于累加法的思想得到数列的通项公式
第三问中,由(Ⅱ)可知,如果
的前
项中恰有
项
取
,
的后
项中恰有
项
取
,则
,可知分析得到结论。
解:(Ⅰ)由题设,满足条件的数列
的所有可能情况有:
(1)
此时
;(2)
此时
;
(3)
此时
;(4)
此时
;
(5)
此时
;(6)
此时
;
所以,
的所有可能的值为:
,
,
,
,
. ……4分
(Ⅱ)由
,
可设
,则
或
(
,
),
因为
,所以 ![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220224539747.png)
.
因为
,所以
,且
为奇数,
是由
个1和
个
构成的数列
所以![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232202249131260.png)
.
则当
的前
项取
,后
项取
时
最大,
此时![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222105568.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232202253351365.png)
.
证明如下:
假设
的前
项中恰有
项
取
,则
的后
项中恰有
项
取
,其中
,
,
,
.
所以![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220225959568.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232202259741943.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232202260831300.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220226115995.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220226130915.png)
.
所以
的最大值为
. ……9分
(Ⅲ)由(Ⅱ)可知,如果
的前
项中恰有
项
取
,
的后
项中恰有
项
取
,则
,若
,则
,因为
是奇数,所以
是奇数,而
是偶数,因此不存在数列
,使得
. ……13分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222433367.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222449497.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222480456.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222495452.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222636641.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222870639.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223073651.png)
第二问中,因为递推关系可知由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220221903646.png)
可设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223119544.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223135409.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223151418.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220221684536.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223275460.png)
那么借助于累加法的思想得到数列的通项公式
第三问中,由(Ⅱ)可知,如果
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223291563.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223307480.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223338267.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223353607.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223369226.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223291563.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223307480.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223338267.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223463588.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223541209.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232202235561443.png)
解:(Ⅰ)由题设,满足条件的数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222433367.png)
(1)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222449497.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223712629.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223743420.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222636641.png)
(3)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222480456.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223790617.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223821518.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222870639.png)
(5)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222495452.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223790617.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220224102438.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223073651.png)
所以,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222090556.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222215248.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222246291.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222261262.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222277302.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222293257.png)
(Ⅱ)由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220221903646.png)
可设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223119544.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223135409.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223151418.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220221684536.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223275460.png)
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220224523529.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220224539747.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220224570765.png)
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220221637506.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220224711655.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220224726277.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223291563.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223307480.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223307480.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220224804224.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232202249131260.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232202249451035.png)
则当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223291563.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223307480.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220225132206.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223307480.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220224804224.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222105568.png)
此时
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222105568.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232202253351365.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220225397625.png)
证明如下:
假设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223291563.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223307480.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223338267.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220225475602.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223369226.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223291563.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223307480.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223338267.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223463588.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223541209.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220225865637.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220225896732.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220225912691.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220225927536.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220225959568.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232202259741943.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232202260831300.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220226115995.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220226130915.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232202261611316.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222105568.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222339612.png)
(Ⅲ)由(Ⅱ)可知,如果
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223291563.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223307480.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223338267.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223353607.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223369226.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223291563.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223307480.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223338267.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223463588.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220223541209.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232202235561443.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222152945.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220226848987.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220224726277.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220226895409.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220226910893.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222121366.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220222355942.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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