题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001718-32664.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001719-68938.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001719-92689.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001719-44600.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001719-30981.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001720-24006.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001720-88052.png)
(1)求
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001720-46552.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001720-84455.png)
(2)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001721-60466.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001721-84279.png)
(3)是否存在唯一正整数
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001721-12851.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001721-30781.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001722-46551.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001722-82404.png)
解析
试题分析:本题考查数列的求值,等比数列的证明和研究不等式的恒成立问题.(1)通过题设条件给出的数列关系,求出数列的初始值;(2)根据等比数列的定义,分别得到证明,其中应说明第一项不为零;(3)探求是否存在唯一的正整数
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001721-12851.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001721-30781.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001722-87853.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001719-68938.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001722-20643.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001723-59560.png)
试题解析:(1)依题意,
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001723-78586.png)
(2)证明:依题意,对任意正整数
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001719-30981.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001723-93999.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001723-56586.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001724-63971.png)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001724-39543.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001724-92052.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001724-96813.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001724-23401.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001725-27746.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001725-20434.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001725-62471.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001724-92052.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001721-84279.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001725-19498.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001726-46784.png)
(3)由(2)得
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001726-85135.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001726-78436.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001718-32664.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001719-68938.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001723-59560.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001726-39474.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001722-20643.png)
又令
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001727-29736.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001727-99417.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001727-53481.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001727-17789.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001728-45970.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001726-39474.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001728-37936.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001728-68256.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001726-39474.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001728-68256.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001728-30249.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001724-92052.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001723-59560.png)
综上所述,存在唯一的正整数
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001723-59560.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001726-39474.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001721-30781.png)
核心考点
试题【已知数列,满足,,且对任意的正整数,和均成等比数列.(1)求、的值;(2)证明:和均成等比数列;(3)是否存在唯一正整数,使得恒成立?证明你的结论.】;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/20191012/20191012001650-35684.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001651-83618.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001651-49444.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001651-37023.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001651-15443.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001651-72250.png)
(Ⅰ)求
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001652-99891.png)
(Ⅱ)若
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001652-62426.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001652-80717.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001652-10192.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001644-44527.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001644-99357.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001644-60802.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001644-52166.png)
A.15 | B.30 | C.31 | D.64 |
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001632-44194.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001632-75375.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001633-98985.png)
A.![]() | B.![]() | C.5 | D.6 |
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001618-67061.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001619-78850.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001619-93353.png)
(Ⅰ)求数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001618-67061.png)
(Ⅱ)若
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001619-81516.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001619-61931.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001620-43402.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001620-47862.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001551-46044.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001551-33317.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001551-61876.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001551-83111.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001552-35239.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001552-80653.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001552-63876.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001552-74596.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012001552-66152.png)
A.5 | B.4 | C.3 | D.2 |
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