已知單調(diào)遞增的等比數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429727481.png)
滿足:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429743662.png)
,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429743456.png)
是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429758344.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429774334.png)
的等差中項.
(1)求數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429727481.png)
的通項公式;
(2)設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429805679.png)
,求數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429821491.png)
的前
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429836277.png)
項和
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429852388.png)
.
(1)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429868475.png)
;(2)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429883842.png)
.
試題分析:(1)先由條件“
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429743456.png)
是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429758344.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429774334.png)
的等差中項”得到
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429930757.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429946652.png)
,然后利用首項
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429961315.png)
和公比
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429977310.png)
將相關(guān)的等式表示,構(gòu)建二元方程組,求出首項
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429961315.png)
和公比
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429977310.png)
的值,從而確定數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429727481.png)
的通項公式;(2)先求出數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429821491.png)
的通項公式,根據(jù)數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429821491.png)
的通項公式選擇錯位相減法求數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429821491.png)
的前
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429836277.png)
項和
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429852388.png)
.
試題解析:(1)由題意知:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429930757.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030430133750.png)
,
又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429743662.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030430164756.png)
,
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030430180467.png)
(不合題意)或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030430180408.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030430195435.png)
, 故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429868475.png)
;
(2)由(1)知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030429868475.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030430242606.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240304302581004.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240304302731265.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240304302891589.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030430304860.png)
.
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