题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073246-31193.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073246-47291.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073246-26044.gif)
(Ⅰ)当
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073247-44748.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073247-89008.gif)
(Ⅱ)若数列
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073247-87092.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073247-49069.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073248-68523.gif)
(Ⅲ)设
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073246-26044.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073246-26044.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073246-26044.gif)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073248-64135.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073248-70276.gif)
(Ⅱ)证明:因为
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073248-26801.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073248-52855.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073249-19724.gif)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073247-49069.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073249-27868.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073249-74400.gif)
若
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073249-27868.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073249-16171.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073250-34142.gif)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073250-28439.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073250-10059.gif)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073250-29084.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073251-58700.gif)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073251-89409.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073251-10767.gif)
即当
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073249-27868.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073251-98777.gif)
若
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073249-74400.gif)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073251-40062.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073252-46226.gif)
所以,当
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073247-49069.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073248-68523.gif)
(Ⅲ)证明:由(Ⅱ)易知对于四个数的数列,若第三项的值介于前两项的值之间,则交换第二项与第三项的位置将使数列波动强度减小或不变.(将此作为引理)
下面来证明当
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073252-79675.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073246-26044.gif)
(ⅰ)证明
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073252-18239.gif)
若
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073252-92615.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073252-57552.gif)
若
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073253-73482.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073253-23809.gif)
所以,
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073253-94844.gif)
(ⅱ)设
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073253-89527.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073254-58557.gif)
若
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073254-83853.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073254-76619.gif)
若
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073254-97627.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073255-52376.gif)
所以,
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073255-40755.gif)
(ⅲ)设
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073255-67904.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073255-69177.gif)
若
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073255-83610.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073256-77629.gif)
则由前面推理可得
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073256-87045.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073255-67904.gif)
所以,
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073255-69177.gif)
综上,得证.
同理可证:当
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073256-58011.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073246-26044.gif)
解析
核心考点
试题【定义为有限项数列的波动强度.(Ⅰ)当时,求;(Ⅱ)若数列满足,求证:;(Ⅲ)设各项均不相等,且交换数列中任何相邻两项的位置,都会使数列的波动强度增加,求证:数列】;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/20191011/20191011073152-27169.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073152-34981.gif)
(1)求数列{an}的通项公式;
(2)记bn=2(1-
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073152-37151.gif)
(3)若cn=
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073153-65990.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073153-13003.gif)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073120-49566.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073120-53991.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073120-23174.png)
A.![]() | B.![]() | C.![]() | D.![]() |
数列
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073112-29781.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073112-40086.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073112-65614.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073113-98672.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073113-89904.gif)
(1)求数列
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073112-29781.gif)
(2)设
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073113-51681.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073113-11641.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073112-65614.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073113-25912.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073114-18646.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073051-53781.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073052-95498.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073052-13018.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073052-14322.gif)
(I)求数列
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073052-58999.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073053-78736.gif)
(Ⅱ)若数列{
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073053-36518.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073053-69369.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073053-47160.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073054-51031.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191011/20191011073054-70601.gif)
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