(本小題滿分12分)
已知數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823195530259481.png)
滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823195530275370.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823195530290932.png)
.
⑴求數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823195530259481.png)
的通項公式;
⑵若數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823195530322491.png)
滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408231955303371235.png)
,求數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823195530322491.png)
的通項公式.
解:(1)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823195530368607.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823195530384735.png)
,
而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823195530275370.png)
,故數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823195530415523.png)
是首項為2,公比為2的等比數(shù)列,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823195530431587.png)
,因此
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823195530446522.png)
. ( 5分)
(2)∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408231955304621175.png)
,∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823195530478878.png)
,( 7分)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408231955304931179.png)
,
即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408231955305091158.png)
,①
當
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823195530524431.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408231955305401560.png)
,②
①-②得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823195530571851.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823195530587875.png)
. 。10分)
可驗證
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823195530602356.png)
也滿足此式,因此
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823195530634665.png)
. 。12分)
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