如果數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052659743449.png)
滿足:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052659775647.png)
且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240526597901198.png)
,則稱數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052659743449.png)
為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052659821277.png)
階“歸化數(shù)列”.
(1)若某4階“歸化數(shù)列”
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052659743449.png)
是等比數(shù)列,寫出該數(shù)列的各項(xiàng);
(2)若某11階“歸化數(shù)列”
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052659743449.png)
是等差數(shù)列,求該數(shù)列的通項(xiàng)公式;
(3)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052659743449.png)
為n階“歸化數(shù)列”,求證:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240526598991070.png)
.
試題分析:(1)等比數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052659993456.png)
是4階“歸化數(shù)列”,則有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700009791.png)
,這樣
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700024395.png)
,于是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700040774.png)
,從而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700071484.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700087493.png)
,以后各項(xiàng)依次可寫出;(2)等差數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052659993456.png)
是11階“歸化數(shù)列”,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700118737.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700133423.png)
,這樣有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700149639.png)
,知當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700165436.png)
時(shí),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700196822.png)
,當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700211430.png)
時(shí),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700227814.png)
,由此可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052659993456.png)
的通項(xiàng)公式分別為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700258715.png)
或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700274724.png)
;(3)對(duì)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700289277.png)
階“歸化數(shù)列”,從已知上我們只能知道在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700305557.png)
中有正有負(fù),因此為了求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700321999.png)
,我們可以設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700352579.png)
是正的,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700367626.png)
是負(fù)的,這樣
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700383785.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700399805.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700321999.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700461956.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700477851.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700492652.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700508548.png)
證畢.
(1)設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700555543.png)
成公比為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700586311.png)
的等比數(shù)列,顯然
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700601419.png)
,則由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700617643.png)
,
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700633856.png)
,解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700648395.png)
,由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700679699.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700695456.png)
,解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700711494.png)
,
所以數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052659915522.png)
或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052659931513.png)
為所求四階“歸化數(shù)列”; 4分
(2)設(shè)等差數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700773600.png)
的公差為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700789295.png)
,由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700804685.png)
,
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700820730.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700851568.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700867423.png)
, 6分
當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700882369.png)
時(shí),與歸化數(shù)列的條件相矛盾,
當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700898390.png)
時(shí),由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700913914.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700945643.png)
,
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240527009601295.png)
8分
當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700976391.png)
時(shí),由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700991899.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052701007653.png)
,
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052701023963.png)
(n∈N
*,n≤11),
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240527010381260.png)
(n∈N
*,n≤11), 10分
(3)由已知可知,必有a
i>0,也必有a
j<0(i,j∈{1,2, ,n,且i≠j).
設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052701054641.png)
為諸a
i中所有大于0的數(shù),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052701085688.png)
為諸a
i中所有小于0的數(shù).
由已知得X=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052701101334.png)
+
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052701116344.png)
+ +
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052701132336.png)
=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052701147320.png)
,Y=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052701163349.png)
+
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052701194356.png)
+ +
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052701210382.png)
=-
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052701147320.png)
.
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052701241858.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240527012571595.png)
. 16分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052700289277.png)
項(xiàng)和公式,不等式的放縮法.
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