题目
题型:不详难度:来源:
设正数数列{an}的前n项和Sn满足
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050143-36306.png)
(1) 求a1的值;
(2) 证明:an=2n-1;
(3) 设
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050143-44063.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050143-36306.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050143-25061.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050144-78284.png)
∴an=Sn-Sn-1=
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050144-41873.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050144-92710.png)
整理得 (an+an-1)(an-an-1-2)=0
∵an>0, ∴an+an-1>0
∴an-an-1-2=0,即an-an-1=2(n≥2).
∴{an}是等差数列,∴an=2n-1.
(3)∵
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050143-44063.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050144-57286.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050145-25376.png)
∴Tn=
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050145-83194.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050145-16657.png)
解析
核心考点
试题【.(本小题满分12分)设正数数列{an}的前n项和Sn满足.(1) 求a1的值;(2) 证明:an=2n-1;(3) 设,记数列{bn}的前n项为Tn,求】;主要考察你对数列综合等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050135-98294.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050135-58728.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050135-45715.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050135-30009.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050135-39994.png)
已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050112-55454.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050112-21472.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050112-11353.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050112-81079.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050112-47668.png)
(Ⅰ)求
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050112-33621.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050112-45671.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050112-67158.png)
(Ⅱ)用数学归纳法证明:当
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050113-27735.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050113-70083.png)
(Ⅲ)证明:当
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050113-27735.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050113-86314.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050105-99382.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050105-57386.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050105-99382.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050105-41952.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050106-66937.png)
各项均为正数的数列
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050054-31423.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050054-55078.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050054-48910.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050054-30926.png)
(1)求数列
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050054-31423.png)
(2)若数列
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050055-13744.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050055-33308.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050055-18882.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050055-60255.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050055-18882.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050054-55078.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050056-93459.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050056-93459.png)
(3)若数列
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050056-88535.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050056-93459.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050056-84194.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050056-76411.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050050-90576.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050050-53770.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191015/20191015050051-17013.png)
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