试题分析:(Ⅰ)已知前
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315191297.png)
项和公式
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315207388.png)
求
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315222348.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240323152381094.png)
.由此可得数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315035415.png)
的通项公式.
(Ⅱ)由等差数列与等比数列的积或商构成的新数列,求和时用错位相消法.在本题中用错位相消法可得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315269669.png)
.这也是一个数列,要求数列的范围,首先考查数列的单调性,而考查数列的单调性,一般是考查相邻两项的差的符号.作差易得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315285801.png)
,所以这是一个递增数列,第一项即为最小值.递增数列有可能无限增大,趋近于无穷大.本题中由于
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315316661.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315332662.png)
.由此即得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315129331.png)
的取值范围.
试题解析:(Ⅰ)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315363334.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315363467.png)
;
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315394418.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240323153941073.png)
,经验证,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315410361.png)
满足上式.
故数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315035415.png)
的通项公式
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315160436.png)
. 4分
(Ⅱ)可知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315456925.png)
,
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240323154721032.png)
,
两式相减,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240323154881313.png)
,
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315269669.png)
. 8分
由于
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315285801.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315129331.png)
单调递增,故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315534601.png)
,
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315550716.png)
,
故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315129331.png)
的取值范围是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032315176471.png)
12分