试题分析:(1)由于64不算大,可以依次计算出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333497488.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333513264.png)
,因为按照定义
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333544409.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333544426.png)
,而此开始
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333560638.png)
,故可得出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333575481.png)
通项公式;(2)显然
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333591315.png)
必须是整数,而且要计算
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333607439.png)
,因此我们可以根据
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333591315.png)
的值分类讨论(分成四类
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333638949.png)
).(3)
要证不等式
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333653807.png)
,最好能求出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333653388.png)
,那么也就要求出数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333575481.png)
的各项,那么我们根据数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333575481.png)
定义,由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333700571.png)
为奇数,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333716777.png)
为偶数,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333731764.png)
为奇数,接下来各项都是偶数,一起到某项为1,下面一项为0,以后全部为0.实际上项为1的项是第
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333747337.png)
项,且
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333747460.png)
时
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333763464.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333778513.png)
时
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333794432.png)
,因此
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333809411.png)
是最大的,但在计算
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333809411.png)
时,要注意当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333841548.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333856598.png)
,只要它不为0,就可继续下去.
试题解析:(1)由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333856588.png)
,可得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333872462.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333903464.png)
,…,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333903453.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333934464.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333965620.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333965417.png)
,…,
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333981415.png)
的前7项成等比数列,从第8起数列的项均为0. (2分)
故数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333981415.png)
的通项公式为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240313334821487.png)
. (4分)
(2)若
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334075668.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334090657.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334106639.png)
,
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334121450.png)
成等差数列,可知即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334137647.png)
,解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334153390.png)
,故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334168391.png)
;(舍去)
若
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334184708.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334215694.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334106639.png)
,
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334121450.png)
成等差数列,可知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334246758.png)
,解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334277367.png)
,故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334324401.png)
;(舍去)(3分)
若
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334340749.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334371718.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334387677.png)
,
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334121450.png)
成等差数列,可知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334433780.png)
,解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334153390.png)
,故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334465405.png)
;
若
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334699735.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334792735.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334387677.png)
,
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334121450.png)
成等差数列,可知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334870799.png)
,解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334277367.png)
,故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334886373.png)
;(舍去)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334901294.png)
的值为2. (6分)
(3)由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334917534.png)
(
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334933453.png)
),可得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333716777.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333731764.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334979789.png)
,
若
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031334995777.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031335011345.png)
是奇数,从而
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031335026973.png)
,
可得当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031335026578.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031335042608.png)
成立. (3分)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031335057636.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031335073503.png)
,…
故当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031335089456.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031335104443.png)
;当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031335120507.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031335135420.png)
. (5分)
故对于给定的
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031335135339.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031335151345.png)
的最大值为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031335167553.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240313351821314.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240313351981152.png)
,
故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031335213749.png)
. (8分)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031333248297.png)
项和与最大值.