试题分析:(1)将等差数列中的相应式子转化为首项和公差的二元一次方程组,求出首项和公差,最后再利用等差数列的通项公式
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832336801.png)
即可求出等差数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832087481.png)
的通项公式;(2)先将数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832118693.png)
的通项公式结构选择裂项求和法求数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832118693.png)
的前
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832149297.png)
项和
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832165388.png)
,然后根据条件列式,利用正整数的一些相关性质列不等式求出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832196337.png)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832149297.png)
的值.
试题解析:(1)设等差数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832087481.png)
的公差为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832477321.png)
,
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832492824.png)
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832508956.png)
2分
解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832523733.png)
3分
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240208325391142.png)
.
所以数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832087481.png)
的通项公式为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832305587.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832586589.png)
. 4分
(2)因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240208326011694.png)
, 5分
所以数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832118693.png)
的前
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832149297.png)
项和
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240208326481334.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240208326793426.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240208327111009.png)
. 7分
假设存在正整数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832196337.png)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832149297.png)
,且
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832227484.png)
,使得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832227343.png)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832243411.png)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832165388.png)
成等比数列,
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832789615.png)
. 8分
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240208328041164.png)
. 9分
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832820929.png)
.
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832835421.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832851732.png)
.
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832851686.png)
.
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832882421.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832882855.png)
.
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832898526.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832305441.png)
. 12分
此时
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240208329291017.png)
. 13分
所以存在满足题意的正整数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832196337.png)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832149297.png)
,且只有一组解,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832305441.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020832321453.png)
. 14分