题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143247-16014.gif)
(Ⅰ)求证:“若数列{an}是等比数列,则数列{bn}也是等比数列”是真命题;
(Ⅱ)写出(Ⅰ)中命题的逆命题;判断它是真命题还是假命题,并说明理由.
答案
(I)证明见解析
(II)逆命题是:若
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143248-49489.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143248-19362.gif)
解析
(I)因为
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143248-19362.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143248-62907.gif)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143248-71136.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143249-79262.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143248-49489.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143249-50460.gif)
(II)(I)中命题的逆命题是:若
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143248-49489.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143248-19362.gif)
……………………………………………………………8分
设
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143248-49489.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143249-26938.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143250-21386.gif)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143250-45057.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143250-62339.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143251-99398.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143251-59976.gif)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143248-19362.gif)
但当q≠a2时,
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143248-19362.gif)
故逆命题是假命题.……………………………………………………………………12分
另解:取a=2,q=1时,
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143251-33469.gif)
因此
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143248-49489.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143248-19362.gif)
故逆命题是假命题.……………………………………………………………………12分
核心考点
试题【已知数列{an}、{bn}满足:a1=1,a2=a(a为实数),且,其中n=1,2,3,…(Ⅰ)求证:“若数列{an}是等比数列,则数列{bn}也是等比数列”是】;主要考察你对四种命题等知识点的理解。[详细]
举一反三
A.命题“若![]() ![]() ![]() ![]() |
B.若命题![]() ![]() ![]() |
C.若![]() ![]() |
D.“![]() ![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143224-35684.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143224-29182.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143224-29182.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143225-34448.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143225-34448.gif)
A.充分不必要条件 | B.必要不充分条件 |
C.充要条件 | D.既不充分也不必要条件 |
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143219-68967.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143220-72773.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143220-20352.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143220-70677.gif)
①若
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143219-68967.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143220-33110.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143221-81702.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143221-81702.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143221-51857.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143220-33110.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143220-72773.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143220-70677.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143221-51857.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143220-33110.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143221-81702.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143219-68967.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143220-33110.gif)
①G2=ab是三个数a、G、b成等比数列的充要条件;②若函数y=f(x)对任意实数x都满足f(x+2)=-f(x),则f(x)是周期函数;③对于命题
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143207-41415.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143207-46291.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143208-62060.gif)
其中不正确命题的序号为_______(把你认为不正确的命题序号都填上).
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143159-99998.gif)
①由线C不可能表示椭圆;
②当1<k<4时,曲线C表示椭圆;
③若曲线C表示双曲线,则k<1或k>4;
④若曲线C表示焦点在x轴上的椭圆,则1<k<
![](http://img.shitiku.com.cn/uploads/allimg/20191102/20191102143159-76288.gif)
其中所有正确命题的序号为_______ ______.
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