题目
题型:四川省月考题难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015014000-20366.gif)
(Ⅰ)求证:xn>3;
(Ⅱ)求证:xn+1<xn;
(Ⅲ)求数列{xn}的通项公式。
答案
1)当n=1时,
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015014000-80858.gif)
2)假设n=k(n≥1)时结论成立,即
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015014000-40383.gif)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015014000-77972.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015014001-32132.gif)
即n=k+1时,结论成立;
由1)2)可知对任意的正整数n,都有
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015014001-71806.gif)
(Ⅱ)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015014001-33320.gif)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015014001-44942.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015014002-82558.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015014002-70045.gif)
(Ⅲ)解:
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015014002-72757.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015014002-99237.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015014003-47102.gif)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015014003-22997.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015014003-35638.gif)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015014003-34278.gif)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015014003-72096.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015014004-15478.gif)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015014004-38358.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015014004-30004.gif)
核心考点
举一反三
(1)记bn=an+n+1,求证:数列{bn}是等比数列,并写出数列{an}的通项公式;
(2)在(1)的条件下,记
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015013946-52054.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015013946-89809.gif)