醉寅
幼苗
共回答了15个问题采纳率:86.7% 举报
已知
![](https://img.yulucn.com/upload/b/85/b85a4c84f39a6491e7e313dc30ce6b83_thumb.jpg)
.经计算得
![](https://img.yulucn.com/upload/0/7c/07c7aec0210691971da591b5503ac4f6_thumb.jpg)
,
![](https://img.yulucn.com/upload/e/73/e7329ddeb517983dc24ba086ae2417f6_thumb.jpg)
,
![](https://img.yulucn.com/upload/c/31/c3182431df7edc913de39c4cca9dcce8_thumb.jpg)
,
![](https://img.yulucn.com/upload/0/2c/02c05315ae9523e1cc7bc99cb2a5b3bc_thumb.jpg)
,
![](https://img.yulucn.com/upload/a/ef/aef06ca5eefa808b686cce606aabde31_thumb.jpg)
,通过观察,我们可以得到一个一般性的结论.
(1)试写出这个一般性的结论;
(2)请用数学归纳法证明这个一般性的结论;
(3)对任一给定的正整数
![](https://img.yulucn.com/upload/8/20/8200869c460c0ab0c99ba0bfdca37010_thumb.jpg)
,试问是否存在正整数
![](https://img.yulucn.com/upload/e/12/e12b456b13ad6fcc87ab6286a911ced0_thumb.jpg)
,使得
![](https://img.yulucn.com/upload/9/f4/9f4a139cdaf2b3eac6ea31c4192a53f5_thumb.jpg)
?
若存在,请给出符合条件的正整数
![](https://img.yulucn.com/upload/e/12/e12b456b13ad6fcc87ab6286a911ced0_thumb.jpg)
的一个值;若不存在,请说明理由.
见解析
(1)观察规律2,4,8,16,…,
![](https://img.yulucn.com/upload/2/59/259860fa2a69551f24762fa219339698_thumb.jpg)
;
![](https://img.yulucn.com/upload/2/8c/28ca4ba89898f23520f5dcff92e8ed47_thumb.jpg)
,所以
![](https://img.yulucn.com/upload/2/1d/21de1dcf98f41b366d32b55227335ee8_thumb.jpg)
.
(2)用数学归纳法证明时要分两个步骤:一是先验证:当n=1时,不等式成立;二是先假设n=k时,不等式成立,再证明当n=k+1时,命题也成立,但一定要用上n=k时的归纳假设.
(3)令
![](https://img.yulucn.com/upload/b/6b/b6b93caa3dff0126d031e96ff1abbfd6_thumb.jpg)
,当n=2a时,
![](https://img.yulucn.com/upload/b/f4/bf46e07d3d57b16f0232a330eef28b49_thumb.jpg)
符合要求.所在存在
(1)
![](https://img.yulucn.com/upload/2/66/2660ec20c7fbf9458f90e34edbf69a15_thumb.jpg)
(当且仅当
![](https://img.yulucn.com/upload/6/51/651529fc383aaaa35f032c6a686eec19_thumb.jpg)
时取等号)………4分
(2)证明:(数学归纳法)
![](https://img.yulucn.com/upload/d/db/ddb92c0698f6a90797c9c6ce23f5a03a_thumb.jpg)
当
![](https://img.yulucn.com/upload/6/51/651529fc383aaaa35f032c6a686eec19_thumb.jpg)
时,显然成立
![](https://img.yulucn.com/upload/0/68/0684fc9fe8fda0a641d44c1494d37758_thumb.jpg)
假设当
![](https://img.yulucn.com/upload/7/04/704b9acb7baf120fe32c63d3f2c0fa01_thumb.jpg)
时成立,即
![](https://img.yulucn.com/upload/2/62/2626c5b82f85185c82c3c35074eaff75_thumb.jpg)
……………………6分
当
![](https://img.yulucn.com/upload/6/15/615b1518b990d17731b386c933d7f25c_thumb.jpg)
时,左边
![](https://img.yulucn.com/upload/a/93/a938d05a6bfdc2ef3807d0beff28e824_thumb.jpg)
右边
即当
![](https://img.yulucn.com/upload/6/15/615b1518b990d17731b386c933d7f25c_thumb.jpg)
时,也成立.………………………10分
由
![](https://img.yulucn.com/upload/0/68/0684fc9fe8fda0a641d44c1494d37758_thumb.jpg)
知,
![](https://img.yulucn.com/upload/2/66/2660ec20c7fbf9458f90e34edbf69a15_thumb.jpg)
成立.…………………………12分
(3)存在……………………………………13分
可取
![](https://img.yulucn.com/upload/9/e9/9e9933dba0043843e6ec4d6b2e571f70_thumb.jpg)
……………………………16分
注:答案不唯一
1年前
10