试题分析:本题考查等差数列与等比数列的概念、通项公式、前n项和公式、数列求和等基础知识,考查运算能力和推理论证能力.第一问,将已知条件中的
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030605979304.png)
用
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030606167349.png)
代替得到新的式子,两式子作差,得出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030606182464.png)
为等差数列,注意需检验
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030606198347.png)
的情况,将
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030606198351.png)
求出代入到已知的第2个式子中,用
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030606213344.png)
代替式子中的
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030605979304.png)
,两式子作差得到
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030606245378.png)
表达式;第二问,将
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030606260488.png)
代入到
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030606276498.png)
中,用错位相减法求和.
试题解析:(1)∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030606291813.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030606291838.png)
两式作差得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030606307659.png)
∴当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030606323433.png)
时,数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030605886494.png)
是等差数列,首项
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030606354350.png)
为3,公差为2,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030606385992.png)
,又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030606401371.png)
符合
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030606120771.png)
4分
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240306064321217.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240306064321278.png)
两式相减得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030606447758.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030606463798.png)
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030606479393.png)
不满足,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240306061351011.png)
6分
(2)设
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240306065101433.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240306065101482.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240306065251813.png)
两式作差得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240306065411729.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240306065572482.png)
所以,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030606135947.png)
..12分