试题分析:本题主要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,考查化归与转化思想,考查思维能力、分析问题与解决问题的能力和计算能力.第一问,利用等差数列的通项公式,前n项和公式将
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231168615.png)
展开,利用等比中项得出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231168557.png)
,再利用通项公式将其展开,两式联立解出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231184315.png)
和
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231199320.png)
,从而得出数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231028456.png)
的通项公式;第二问,将第一问的结论代入,再利用等比数列的定义证明数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231230558.png)
是等比数列,利用分组求和法,求出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231230373.png)
的值.
试题解析:(Ⅰ)设等差数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231028456.png)
的公差为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231293416.png)
.
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231168615.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240322313241007.png)
. ①
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231074509.png)
成等比数列,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231355921.png)
. ② 2分
由①,②可得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231371611.png)
. 4分
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231137581.png)
. 6分
(Ⅱ)由题意
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231402607.png)
,设数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231418476.png)
的前
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231418297.png)
项和为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231433373.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231449546.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240322314641324.png)
,所以数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231480435.png)
为以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231496292.png)
为首项,以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032231511248.png)
为公比的等比数列 9分
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240322315271185.png)
12分