题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020503-98154.gif)
若一个数列各项取倒数后按原来的顺序构成等差数列,则称这个数列为调和数列.已知数列
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020504-95615.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020504-26499.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020504-82026.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020504-77116.gif)
(Ⅰ)求证:数列
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020504-26499.gif)
(Ⅱ)把数列
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020504-26499.gif)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020505-41955.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020505-63766.gif)
(Ⅲ)对于(Ⅱ)中的数列
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020504-26499.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020505-72861.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020505-83231.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020505-72861.gif)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020504-82026.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020504-26499.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020506-31491.gif)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020506-58799.gif)
因为数列
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020504-95615.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020506-79080.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020506-64594.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020507-91466.gif)
由①得
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020507-80305.gif)
代入②式得
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020507-89630.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020507-69595.gif)
故
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020508-38499.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020504-26499.gif)
(Ⅱ)设
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020504-26499.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020508-23711.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020508-41534.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020508-80734.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020509-96713.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020509-70317.gif)
于是
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020509-88491.gif)
注意到第
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020509-78567.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020509-77682.gif)
所以三角形数表中第1行至第
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020510-46471.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020510-31484.gif)
因此第
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020505-63766.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020504-26499.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020510-58853.gif)
故第
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020505-63766.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020510-32439.gif)
所以第
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020505-63766.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020511-60033.gif)
(Ⅲ)由
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020511-57008.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020511-28054.gif)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020511-76532.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020511-39034.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020512-95171.gif)
②—① 得
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020512-18488.gif)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020512-92800.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020512-68033.gif)
④-③ 得
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020512-91719.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020513-24291.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020505-72861.gif)
解析
核心考点
试题【(本小题满分14分)若一个数列各项取倒数后按原来的顺序构成等差数列,则称这个数列为调和数列.已知数列是调和数列,对于各项都是正数的数列,满足.(Ⅰ)求证:数列是】;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/20191013/20191013020453-18283.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020454-15154.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020454-71016.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020454-80375.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020454-35191.gif)
A.![]() | B.![]() | C.![]() | D.![]() |
已知数列
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020446-33705.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020446-14568.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020447-68463.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020447-83132.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020447-96098.gif)
(Ⅰ)求证:{
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020447-15303.gif)
(Ⅱ)求数列
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020446-33705.gif)
(Ⅲ)若
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020447-39373.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020448-61821.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020439-47312.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020439-24740.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020439-58840.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020439-47312.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020439-19154.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020440-77395.gif)
设集合W由满足下列两个条件的数列
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020418-58900.gif)
①
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020418-33897.gif)
②存在实数M,使
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020418-56607.gif)
(I)在只有5项的有限数列
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020419-55138.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020419-16452.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020419-24920.gif)
(II)设
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020419-68002.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020419-38886.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020420-80976.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020420-52673.gif)
(III)设数列
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020420-36379.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020420-67004.gif)
求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020421-43171.gif)
已知数列
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020402-52479.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020402-21247.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020402-58042.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020403-17721.gif)
数列
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020403-19943.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020403-57508.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020403-43028.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020404-22089.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020404-28059.gif)
(Ⅰ)求数列
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020402-52479.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020403-19943.gif)
(Ⅱ)设
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020404-47061.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020404-63835.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020402-21247.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020405-68222.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020405-25248.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020403-43028.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020405-54324.gif)
(Ⅲ)设
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020405-35879.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020405-25146.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020406-84652.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013020406-92013.gif)
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