設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358047456.png)
是公比大于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358063206.png)
的等比數(shù)列,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358078388.png)
為數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358047456.png)
的前
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358110297.png)
項(xiàng)和.已知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358125463.png)
,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358141429.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358156418.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358156435.png)
構(gòu)成等差數(shù)列.
(1)求數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358047456.png)
的通項(xiàng)公式;
(2)令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358219903.png)
求數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358234471.png)
的前
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358110297.png)
項(xiàng)和
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358266373.png)
.
(1)數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358047456.png)
的通項(xiàng)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358281501.png)
.(2)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358312928.png)
.
試題分析:(1)設(shè)數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358047456.png)
的公比為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358328514.png)
,
根據(jù)題意建立
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358344435.png)
的方程組,求解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358359695.png)
,從而得出數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358047456.png)
的通項(xiàng)公式.
(2)由(1)得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358390562.png)
, 通過研究
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358406704.png)
,
知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358234471.png)
是以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358422556.png)
為首項(xiàng),以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358437454.png)
為公差的等差數(shù)列,
故可利用等差數(shù)列的求和公式,計算得到
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358312928.png)
.
試題解析:(1)設(shè)數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358047456.png)
的公比為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358328514.png)
,
由已知,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240423585001420.png)
, 2分
即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240423585151019.png)
, 也即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240423585311249.png)
解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358359695.png)
4分
故數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358047456.png)
的通項(xiàng)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358281501.png)
. 6分
(2)由(1)得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358390562.png)
,
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358609976.png)
, 8分
又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358406704.png)
,
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358234471.png)
是以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358422556.png)
為首項(xiàng),以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358437454.png)
為公差的等差數(shù)列 10分
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358671717.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358702887.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358718882.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358734831.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042358312928.png)
. 12分
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