試題分析:(Ⅰ)依次代入寫出27,9,3;8,9,3;6,2,3.
(Ⅱ)根據(jù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908569925.png)
及
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240239085531437.png)
須討論
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908771345.png)
被3除余1,,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908771345.png)
被3除余2,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908771345.png)
被3除余0,等三種情況.
(Ⅲ)注意由已知遞推關系推得數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908834454.png)
滿足:
當
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908849690.png)
時,總有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908865494.png)
成立,其中
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908881673.png)
.
因此應注意討論當
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908896610.png)
時,數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908834454.png)
中大于3的各項:
按逆序排列各項,構成的數(shù)列記為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908927461.png)
,由(Ⅰ)可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908943418.png)
或9,
由(Ⅱ)的證明過程即可知數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908927461.png)
的項滿足:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908990510.png)
,且當
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908990356.png)
是3的倍數(shù)時,若使
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909005498.png)
最小,需使
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909021693.png)
,
滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909005498.png)
最小的數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908927461.png)
中,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909068430.png)
或7,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909083628.png)
,
得到數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909099566.png)
是首項為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909115295.png)
或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909130289.png)
的公比為3的等比數(shù)列,應用等比數(shù)列的通項公式即可得出結論.
解答本題的關鍵是注意“轉化”成等比數(shù)列問題.
試題解析:(Ⅰ)27,9,3;8,9,3;6,2,3. 3分
(Ⅱ)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908771345.png)
被3除余1,則由已知可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909146525.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909161976.png)
;
若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908771345.png)
被3除余2,則由已知可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909146525.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909208740.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909224815.png)
;
若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908771345.png)
被3除余0,則由已知可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909255597.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909255723.png)
;
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909255723.png)
,
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240239092861125.png)
所以,對于數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908491426.png)
中的任意一項
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908771345.png)
,“若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909505453.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909520505.png)
”.
因為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909536505.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909551561.png)
.
所以數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908491426.png)
中必存在某一項
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909583500.png)
(否則會與上述結論矛盾�。�
若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909598473.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909614686.png)
;若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909629477.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909661685.png)
,若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909676440.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909692679.png)
,
由遞推關系易得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908631632.png)
. 8分
(Ⅲ)集合
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908678298.png)
中元素個數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908693672.png)
的最大值為21.
由已知遞推關系可推得數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908834454.png)
滿足:
當
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908849690.png)
時,總有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908865494.png)
成立,其中
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908881673.png)
.
下面考慮當
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908896610.png)
時,數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908834454.png)
中大于3的各項:
按逆序排列各項,構成的數(shù)列記為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908927461.png)
,由(I)可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908943418.png)
或9,
由(Ⅱ)的證明過程可知數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908927461.png)
的項滿足:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908990510.png)
,且當
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908990356.png)
是3的倍數(shù)時,若使
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909005498.png)
最小,需使
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909021693.png)
,
所以,滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909005498.png)
最小的數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908927461.png)
中,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909068430.png)
或7,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909083628.png)
,
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023910097831.png)
,所以數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909099566.png)
是首項為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909115295.png)
或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023909130289.png)
的公比為3的等比數(shù)列,
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023910160753.png)
或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023910175741.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023910175551.png)
或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023910207628.png)
,
因為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023910207655.png)
,所以,當
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908647522.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023910238307.png)
的最大值是6,
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023910253489.png)
,所以集合
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908709298.png)
重元素個數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824023908693672.png)
的最大值為21. 13分