题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000142-38635.png)
(1)证明:数列{an}为等差数列的充要条件是α+2β=0;
(2)已知α=1,β=
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000142-80109.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000142-90088.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000143-96544.png)
答案
(2)由已知得
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000143-59370.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000143-92934.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000143-68532.png)
解析
试题分析:(1)解:已知数列
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000145-44975.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000145-73345.png)
①充分性:若
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000145-91402.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000146-96670.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000146-11671.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000145-44975.png)
②必要性:若
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000145-44975.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000146-93464.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000145-73345.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000146-47120.png)
化简得
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000147-80342.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000147-85647.png)
因此,数列{an}为等差数列的充要条件是α+2β=0. 8分
(2)由已知得
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000143-59370.png)
又因为
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000147-86956.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000143-92934.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000147-79928.png)
从而有n≥2时,
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000148-51625.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000148-64084.png)
于是由上述两式,得
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000143-68532.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000148-96774.png)
由指数函数的单调性可知,对于任意n≥2,| an+1-an-1|=
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000148-56129.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000148-97022.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000148-56129.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000149-53321.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000148-56129.png)
所以,数列
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000149-59266.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000148-56129.png)
而对于任意的n≥1时,n+
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000143-96544.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000143-96544.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000148-56129.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000143-96544.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000148-56129.png)
因此,数列
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000149-59266.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010000143-96544.png)
16分
点评:解决的关键是对于概念的准确运用,以及利用函数的性质来证明数列之间的关系。属于中档题。
核心考点
试题【设非常数数列{an}满足an+2=,n∈N*,其中常数α,β均为非零实数,且α+β≠0.(1)证明:数列{an}为等差数列的充要条件是α+2β=0;(2)已知α】;主要考察你对等差数列等知识点的理解。[详细]
举一反三
(1)求S5,S7的值;
(2)求证:对任意n∈N*,Sn≥0.
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235839-92301.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235839-75978.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235839-57751.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235839-41284.png)
(Ⅰ) 若
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235840-36722.png)
(i)求
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235840-55883.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235839-92301.png)
(ii)求数列
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235840-61715.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235840-43906.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235841-41010.png)
(Ⅱ)若
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235841-13784.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235841-71087.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235841-98147.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235841-75324.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235833-69273.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235833-13797.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235833-21614.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235833-68877.png)
A.2 | B.3 | C.-3 | D.4 |
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235815-44729.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235815-54868.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235815-81872.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235816-10844.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235816-35076.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235816-87561.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235816-25222.png)
(Ⅰ)求证:数列
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235817-59292.png)
(Ⅱ)求数列
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235816-87561.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235817-44553.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235817-35441.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235803-89713.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235804-17129.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235804-39247.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235804-43690.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235804-64288.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235805-40143.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235805-57045.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009235805-83045.png)
A.5 | B.4 | C.3 | D.2 |
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