题目内容
本题满分14分)
在数列
中,
,且
.
(Ⅰ) 求
,猜想
的表达式,并加以证明;
(Ⅱ) 设
,求证:对任意的自然数
,都有
;
在数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801492480.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801508599.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232218015231067.png)
(Ⅰ) 求
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801539437.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801555348.png)
(Ⅱ) 设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801570876.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801586541.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801617955.png)
解:(1)容易求得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801633494.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801648494.png)
猜想
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801679685.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801695529.png)
(2)见解析.
本试题主要是考查了数列的归纳猜想的思想的运用,以及运用哦递推关系式来求解数列的前几项,并且能运用数学归纳法加以证明,同时对于构造的新数列也能利用裂项法求和的综合运用。
(1)利用递推关系,对于n赋值分别得到前几项,并猜想其通项公式,运用数学归纳法加以证明
(2)根据上一问的结论,表示新数列的通项公式,然后利用裂项的思想求和并证明不等式问题。
解:(1)容易求得:
,
----------------------(2分)
故可以猜想
,
下面利用数学归纳法加以证明:
(i) 显然当
时,结论成立,-----------------(3分)
(ii) 假设当
;
时(也可以
),结论也成立,即
,
--------------------------(4分)
那么当
时,由题设与归纳假设可知:
------------(6分)
即当
时,结论也成立,综上,对
,
成立。--------(7分)
(2)
---(9分)
所以
---------(11分)
所以只需要证明![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221802101937.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221802116256.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221802147641.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221802116256.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221802179703.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221802116256.png)
(显然成立)
所以对任意的自然数
,都有
-------(14分)
(1)利用递推关系,对于n赋值分别得到前几项,并猜想其通项公式,运用数学归纳法加以证明
(2)根据上一问的结论,表示新数列的通项公式,然后利用裂项的思想求和并证明不等式问题。
解:(1)容易求得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801633494.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801648494.png)
故可以猜想
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801679685.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801695529.png)
(i) 显然当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801804588.png)
(ii) 假设当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801820412.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801851427.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801867394.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801882684.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801898528.png)
那么当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801913466.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232218019293635.png)
即当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801913466.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221802023603.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801679685.png)
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232218020542703.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232218020853269.png)
所以只需要证明
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221802101937.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221802116256.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221802147641.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221802116256.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221802179703.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221802116256.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221802225600.png)
所以对任意的自然数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221801695529.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823221802257931.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
练习册系列答案
相关题目