2019高二下·全国·专题练习
1 . 对于不等式
<n+1(n∈N*),某同学用数学归纳法证明的主要过程如下:
(1)当n=1时,
<1+1 ,不等式成立;
(2)假设当n=k(k∈N*)时,不等式成立,有
<k+1,即k2+k<(k+1)2,则当n=k+1时,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a302257faaf216c39819d79b869c7233.png)
=
<
=
=(k+1)+1,所以当n=k+1时,不等式也成立.
则下列说法中正确的有_____________ . (填出所有正确说法的序号)
①证明过程全部正确;②n=1的验证不正确;③n=k的归纳假设不正确;④从n=k到n=k+1的推理不正确.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724c1777008ea406b39b467c3137d24f.png)
(1)当n=1时,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af09066970c5b676e38185230a1a3f18.png)
(2)假设当n=k(k∈N*)时,不等式成立,有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5016ff719ec8a7dabc256543207b7858.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a302257faaf216c39819d79b869c7233.png)
=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc0c664271bd507465c234ac71b64c7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad26aa655a18525316d38c6effb9810f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab6b06e071dc81e93df97b29bcdf9c1e.png)
则下列说法中正确的有
①证明过程全部正确;②n=1的验证不正确;③n=k的归纳假设不正确;④从n=k到n=k+1的推理不正确.
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2 . 已知数列
满足
,且
(
为正整数),利用数列的递推公式猜想数列
的通项公式为
.下面是用数学归纳法的证明过程:
(1)当
时,
满足
,命题成立;
(2)假设
(
为正整数)时命题成立,即
成立,则当
时,由
得
,即
是以
为首项,1为公差的等差数列,所以
,即
,所以
,命题也成立.由(1)(2)知,
.
判断以下评述:( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d379316dbb3a43e30f166865e0b2b9e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b648c181bcdc3d2615ce69dfdaba838.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c87b351f16728b0023fd63678f8103c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b648c181bcdc3d2615ce69dfdaba838.png)
(2)假设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ef7ca2b3e8061384501f668e59696a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3e2ee54ed7432592b537ac501a38f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63ba21f3d0cfc86d40e2e06446623ce0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d379316dbb3a43e30f166865e0b2b9e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3467643d35475b81e68113b368e6f869.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41cf1da18d91f7c98086553d157d1a87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2239ce26f47c223c00954290dd3aa85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6729f8558726a382e15eb3f147df2f03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b648c181bcdc3d2615ce69dfdaba838.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1a193f04226baa51605ca3de04c35c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b648c181bcdc3d2615ce69dfdaba838.png)
判断以下评述:( )
A.猜想正确,推理(1)正确 | B.猜想不正确 |
C.猜想正确,推理(1)(2)都正确 | D.猜想正确,推理(1)正确,推理(2)不正确 |
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3 . 在数学归纳法证明等式“
”时,某学生证明如下:(ⅰ)当
时,左边
,右边
,
原等式成立;(ⅱ)假设
时等式成立,即
,那么当
时,
,即当
时,等式也成立.根据(ⅰ)、(ⅱ)可以判断,等式对任意
都成立.评价该学生的证明情况:______ (选填“正确”或“错误”).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/970669da5005b273c26df0fc6f6b049f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c87b351f16728b0023fd63678f8103c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f90bae886c8ab958aa4c693bf8e0627d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b4e2459d0cf0a99b211d6f197b71004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ef7ca2b3e8061384501f668e59696a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afb5e65c7be63c3454a752869cfe56fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63ba21f3d0cfc86d40e2e06446623ce0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38d9155e51b41ffef5556ecd5a9e1885.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63ba21f3d0cfc86d40e2e06446623ce0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36b98ef143f8159f3a7dafa1fd2f2370.png)
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2019-11-09更新
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87次组卷
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2卷引用:沪教版 高二年级第一学期 领航者 第七章 7.4数学归纳法