已知正項等差數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030534951481.png)
的前
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030534967297.png)
項和為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030534982388.png)
,若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030534998507.png)
,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535013605.png)
成等比數(shù)列.
(Ⅰ)求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030534951481.png)
的通項公式;
(Ⅱ)記
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535045594.png)
的前
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030534967297.png)
項和為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535076373.png)
,求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535076373.png)
.
(Ⅰ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535107589.png)
;(Ⅱ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535138964.png)
試題分析:(Ⅰ)由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535154563.png)
和
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535169624.png)
可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535185538.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535201429.png)
;又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535201389.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535247344.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535263406.png)
成等比數(shù)列,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535279716.png)
,綜合起來可求得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535279460.png)
即可.(Ⅱ)由已知可求出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240305352941121.png)
,即數(shù)列{
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535310365.png)
}是由等差數(shù)列和等比數(shù)列組合而成,前
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030534967297.png)
項和為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535076373.png)
可由錯位相減法求得.
試題解析:(Ⅰ)∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030534998507.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535169624.png)
,∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535185538.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535201429.png)
, 2分
又∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535201389.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535247344.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535263406.png)
成等比數(shù)列,
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535279716.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535450912.png)
, 4分
解得,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535466422.png)
或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535481412.png)
(舍去),
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535497577.png)
,故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535107589.png)
; 6分
(Ⅱ)法1:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240305352941121.png)
,
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240305355441455.png)
, ①
①
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535544384.png)
得,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240305355592190.png)
②
①
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030535575169.png)
②得,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240305355911950.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240305356062760.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240305356221661.png)
. 12分
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