(本題滿分18分)本題共有3個(gè)小題,第1小題滿分4分,第2小題滿分8分,第3小題滿分6分.
已知負(fù)數(shù)

和正數(shù)

,且對任意的正整數(shù)
n,當(dāng)

≥0時(shí), 有[

,

]=
[

,

];當(dāng)

<0時(shí), 有[

,

]= [

,


].
(1)求證數(shù)列{

}是等比數(shù)列;
(2)若

,求證


;
(3)是否存在

,使得數(shù)列

為常數(shù)數(shù)列?請說明理由
(1)當(dāng)≥0時(shí),
bn+1-an+1= -
an= ;
當(dāng)<0,
bn+1-an+1=
bn-
= .
所以,總有
bn+1-an+1= (
bn-
an),
又

,可得

,
所以數(shù)列{
bn-an}是等比數(shù)

列. ………………4分
(2)①由

,可得

,故有

,
∴

,

,從而

,
故當(dāng)
n=1時(shí),

成立. ………………6分
②假設(shè)當(dāng)

時(shí),

成立,即

,
由

,可得

,

, 故有

,
∴

, ………………9分

,故有

∴

,

,故

∴當(dāng)

時(shí),

成立.
綜合①②可得對一切正整數(shù)
n,都有

. ………………12分
(3)假設(shè)存在

,使得數(shù)列

為常數(shù)數(shù)列,
由(1)可得
bn-an=

()
n-1,又

,
故
bn=

()
n-1, ………………14分
由

恒成立,可知≥0,即

()
n ≥0恒成立,
即2
n≤

對任意的正整數(shù)
n恒成立, ………………16分
又

是正數(shù),故
n≤

對任意的正整數(shù)
n恒成立,
因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823170416658519.gif" style="vertical-align:middle;" />是常數(shù),故
n≤

不可能對任意正整數(shù)
n恒成立.
故不存在

,使得數(shù)列

為常數(shù)數(shù)列. ………………18分
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
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
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
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
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
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
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
;
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
,

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
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
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
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
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
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
項(xiàng)和為

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
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
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
取最小值時(shí),

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項(xiàng)和為
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,且
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,
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,則數(shù)列
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的通項(xiàng)公式為、
( )
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科目:高中數(shù)學(xué)
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題型:填空題
數(shù)列
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的通項(xiàng)公式為
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,
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達(dá)到最小時(shí),
n等于_______________.
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科目:高中數(shù)學(xué)
來源:不詳
題型:單選題
正項(xiàng)數(shù)列
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的前n項(xiàng)的乘積
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,則數(shù)列
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的前n項(xiàng)和
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中的最大值是 ( )
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