题目内容
已知数列
是首项为
的等比数列,且满足![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950130679.png)
.
(1) 求常数
的值和数列
的通项公式;
(2) 若抽去数列
中的第一项、第四项、第七项、……、第
项、……,余下的项按原来的顺序组成一个新的数列
,试写出数列
的通项公式;
(3) 在(2)的条件下,设数列
的前
项和为
.是否存在正整数
,使得
?若存在,试求所有满足条件的正整数
的值;若不存在,请说明理由.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950099480.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950114291.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950130679.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950161597.png)
(1) 求常数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950177313.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950099480.png)
(2) 若抽去数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950099480.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950302467.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950348487.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950348487.png)
(3) 在(2)的条件下,设数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950348487.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950411297.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950426373.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950411297.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950520665.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950411297.png)
(1)所求常数
的值为1且
(2)
.
(3)
时,![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950723602.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950177313.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950582459.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232159506921230.png)
(3)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950707364.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950723602.png)
第一问中解:由
得
,,
又因为存在常数p使得数列
为等比数列,
则
即
,所以p=1
故数列
为首项是2,公比为2的等比数列,即
.
此时
也满足,则所求常数
的值为1且![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950582459.png)
第二问中,解:由等比数列的性质得:
(i)当
时,
;
(ii) 当
时,
,
所以![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232159506921230.png)
第三问假设存在正整数n满足条件,则
,
则(i)当
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232159515032081.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950879732.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950910845.png)
又因为存在常数p使得数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950988441.png)
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215951004485.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215951050854.png)
故数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950988441.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950582459.png)
此时
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215951128510.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950177313.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215950582459.png)
第二问中,解:由等比数列的性质得:
(i)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215951300747.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215951316624.png)
(ii) 当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215951347789.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215951362654.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232159506921230.png)
第三问假设存在正整数n满足条件,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232159514091412.png)
则(i)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215951300747.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232159515032081.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215951596395.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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