题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032516-63729.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032516-60142.png)
(1)计算
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032526-83812.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032526-65363.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032526-53113.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032526-24798.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032526-49085.png)
(2)若数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032526-96258.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032527-21058.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032527-66512.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032527-41743.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032527-95375.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032527-62505.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032528-59411.png)
(2)运用数学归纳法证明来分为两步骤来加以证明即可。
解析
试题分析:解:(1)当n=1时,a1=S1=2-a1,∴a1=1.
当n=2时,a1+a2=S2=2×2-a2,∴a2=
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032527-41743.png)
当n=3时,a1+a2+a3=S3=2×3-a3,∴a3=
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032527-95375.png)
当n=4时,a1+a2+a3+a4=S4=2×4-a4,∴a4=
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032527-62505.png)
由此猜想an=
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032528-59411.png)
现用数学归纳法证明如下:
①当n=1时, a1=
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032528-30363.png)
②假设n=k(k≥1且k∈N*)时,结论成立,即ak=
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032528-99800.png)
ak+1=Sk+1-Sk=2(k+1)-ak+1-2k+ak=2+ak-ak+1,
∴2ak+1=2+ak,∴ak+1=
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032528-47125.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032529-23732.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032529-84666.png)
由①②知猜想an=
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032528-59411.png)
(2)由(1)知,
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032529-99387.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032529-78051.png)
解法1:当
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032529-64920.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032530-11942.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032530-14118.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032530-64740.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032530-35448.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032530-33919.png)
解法2:当
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032531-97212.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032531-94925.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032531-25791.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032531-88302.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032532-58655.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032530-33919.png)
解法3: 当
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032529-64920.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032532-75934.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032532-72115.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032532-24019.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032532-29047.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032533-86913.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032533-63104.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032533-25305.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032533-97972.png)
点评:主要是考查了数列的猜想以及数学归纳法的运用,属于基础题。
核心考点
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032354-37957.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032354-46976.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032355-39057.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032355-51569.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032355-96145.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032355-73334.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032355-92374.png)
(1) 求数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032356-69654.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032356-98805.png)
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032356-11011.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032407-11072.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032345-33485.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032345-60017.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032345-84370.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032346-84454.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032346-24424.png)
(1)求数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032345-33485.png)
(2)设数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032346-89690.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032346-88114.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032346-89690.png)
(3)求数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032346-89690.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032345-60017.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032346-38795.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032328-77630.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032328-66683.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032328-16469.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032329-20729.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032309-15855.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032309-26773.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032310-35545.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032310-81070.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032310-38571.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032311-52265.png)
(1)求数列
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032309-15855.png)
(2)令
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032311-28482.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032311-52265.png)
①当
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032309-26773.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032311-54627.png)
②若对一切正整数
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032309-26773.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032311-56743.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032311-79418.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032246-29017.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191012/20191012032251-81180.png)
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