题目
题型:不详难度:来源:
已知数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102409-63639.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102409-32725.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102410-24607.png)
(Ⅰ)求
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102410-22151.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102410-70087.png)
(Ⅱ)求数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102409-63639.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102409-63639.png)
(Ⅲ)如果数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102410-35217.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102411-69183.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102410-35217.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102409-32725.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102411-59146.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102411-84292.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102412-30562.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102412-95813.png)
解析
(Ⅰ)对于n赋值为1,2,得到首项和第二项的值。
(Ⅱ)根据第一问中前两项,可以归纳猜想也可以通过当
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102412-20049.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102412-99041.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102413-83570.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102409-63639.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102409-63639.png)
(Ⅲ)由已知得
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102413-62627.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102413-74722.png)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102413-62577.png)
然后借助于等比数列的通项公式求和得到结论。
解:(Ⅰ)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102411-84292.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102414-38864.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102412-30562.png)
(Ⅱ)当
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102412-20049.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102414-48139.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102413-83570.png)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102414-57620.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102414-68060.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102414-20693.png)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102415-87133.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102415-41486.png)
∴数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102415-91649.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102415-23076.png)
(Ⅲ)由已知得
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102413-62627.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102413-74722.png)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102413-62577.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102413-74722.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102416-76111.png)
∴数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102416-29437.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102416-90711.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102416-63459.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102412-95813.png)
核心考点
试题【 (本小题满分10分)已知数列,其前项和为.(Ⅰ)求,;(Ⅱ)求数列的通项公式,并证明数列是等差数列;(Ⅲ)如果数列满足,请证明数列是等比数列,并求其前项和.】;主要考察你对等差数列等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102403-44131.png)
A.11 | B.19 | C.20 | D.21 |
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102351-44493.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102351-70108.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102351-36933.png)
(1)证明数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102351-52364.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102351-70108.png)
(2)记
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102351-37009.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102352-73605.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102342-47620.png)
(1)求数列{an}的通项公式;
(2)当λ=4时,是否存在互不相同的正整数r,s,t,使得ar,as,at成等比数列?若存在,给出r,s,t满足的条件;若不存在,说明理由;
(3)设Sn为数列{an}的前n项和.若对任意n∈N*,都有(1-λ)Sn+λan≥2λn恒成立,求实数λ的取值范围.
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102330-69876.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102330-50354.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102330-53365.png)
(1)证明:数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102330-43395.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102330-50354.png)
(2)设
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102331-59438.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102331-46505.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102319-36648.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102320-29314.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102320-67527.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102320-65901.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102320-56850.png)
(1)求
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102320-38789.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102321-20650.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102319-36648.png)
(2)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012102321-18607.png)
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