题目内容
. (本小题满分12分)已知数列
满足
,
,![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423621492.gif)
,设
.
(1)求数列
、
的通项公式;
(2)记数列
的前
项和
,求使得
成立的最小整数![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423730192.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423574263.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423589244.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423605258.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423621492.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423636433.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423667444.gif)
(1)求数列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423683263.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423574263.gif)
(2)记数列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423574263.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423730192.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423745220.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423761450.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423730192.gif)
(1)由
,得![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423808606.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423823447.gif)
∴数列
是以
为首项,公比为2的等比数列,![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423886581.gif)
∴
∴
时,
,…,
,
,
累加得![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423979876.gif)
∴
(当
时,也满足) …………6分
(2)由(1)利用分组求和法得
…………9分
,得
,即
,∴![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184424151241.gif)
∴使得
成立的最小整数
. …………12分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423792545.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423808606.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423823447.gif)
∴数列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423683263.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423855403.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423886581.gif)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423901479.gif)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423917244.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423933497.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423948436.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423855403.gif)
累加得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423979876.gif)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423995456.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184424057232.gif)
(2)由(1)利用分组求和法得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184424057896.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184424089693.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184424104420.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184424135419.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184424151241.gif)
∴使得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184423761450.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823184424182138.gif)
略
![](http://thumb.zyjl.cn/images/loading.gif)
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