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1、3 3.2 2数学归纳法的应用目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理1.进一步掌握利用数学归纳法证明不等式的方法和技巧.2.了解贝努利不等式,并能利用它证明简单的不等式.目标导航DIANLITOUXI典例透析SUITANG
2、YANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理1.用数学归纳法证明不等式运用数学归纳法证明不等式的两个步骤实际上是分别证明两个不等式.尤其是第二步:一方面需要我们充分利用归纳假设提供的“便利”,另一方面还需要结合运用比较法、综合法、分析法、反证法和放缩法等其他不等式的证明方法.名师
3、点拨从“n=k”到“n=k+1”的方法与技巧:在用数学归纳法证明不等式的问题中,从“n=k”到“n=k+1”的过渡,利用归纳假设是比较困难的一步,它不像用数学归纳法证明恒等式问题一样,只需拼凑出所需要的结构来,而证明不等式的第二步中,从“n=k”到“n=k+1”,只用拼凑的方法,有时也行不通,因为对不等式来说,它还涉及“放缩”的问题,它可能需要通过“放大”或“缩小”的过程,才能利用上归纳假设,因此,我们可以利用“比较法”“综合法”“分析法”等来分析从“n=k”到“n=k+1”的变化,从中找到“放缩尺度”,准确地拼凑出所需要的结构.目标导航DIANLITOUXI典例透析SUITANGYANLIA
4、N随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理【做一做1-1】 设f(k)是定义在正整数集上的函数,且f(k)满足:“当f(k)k2成立时,总可推出f(k+1)(k+1)2成立.”那么下列命题总成立的是()A.若f(3)9成立,则当k1时,均有f(k)k2成立B.若f(5)25成立,则当k5时
5、,均有f(k)k2成立C.若f(7)49成立,则当k8时,均有f(k)42,对于任意的k4,总有f(k)k2成立.答案:D目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLI
6、AN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理2.贝努利不等式对任何实数x-1和任何正整数n,有(1+x)n1+nx.【做一做2】 设nN+,求证:3n2n.分析:利用贝努利不等式来证明.证明:3n=(1+2)n,根据贝努利不等式,有(1+2)n1+n2=1+2n.上式右边舍去1,得(1+2
7、)n2n.3n2n成立.目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理题型一题型二题型三题型一 用数学归纳法证明不等式 分析:利用数学归纳法证明. 目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHIS
8、HISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理题型一题型二题型三反思在利用数学归纳法证明不等式时,要注意对式子的变形,通过放缩、比较、分析、综合等证明不等式的方法,得出要证明的目标不等式.目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLI
9、TOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理题型一题型二题型三目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHIS
10、HISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理题型一题型二题型三题型二 利用贝努利不等式证明不等式 分析:用求商比较法证明an+14ab2ab+6a2b2=14a2b2=224,g(24)=44-25=224,有f(4)g(24).由此猜想当1n2(nN+)时,f(n)=g(2n).当n3(nN+)时,f(n)g(2n).下面用数学归纳法证明.(1)当n=3时,由上述计算知猜想成立;目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI
11、典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理题型一题型二题型三(2)假设当n=k(k3,kN+)时,猜想成立,则f(k)g(2k),即(a+b)k-ak-bk4k-2k+1,则当n=k+1时,f(k+1)=(a+b)k+1-ak+1-bk+1=(a+b)(a+b)k-aak-bbk=(a+b)(a+b)k-ak-bk+akb+abk,反思利用数学归纳法解决探索
12、型不等式问题的思路是:先通过观察、判断、猜想得出结论,然后用数学归纳法证明结论.目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理题型一题型二题型三目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHI
13、SHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理题型一题型二题型三目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUI
14、TANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理12345答案:C 目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理
15、12345答案:B 目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理12345答案:A 目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析S
16、UITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理12345目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGYANLIAN随堂演练ZHISHISHULI知识梳理123455已知acdb0,a+b=c+d,n为大于1的正整数,求证:an+bncn+dn.
高中数学 第二章 几个重要的不等式 2.3 数学归纳法与贝努利不等式 2.3.2 数学归纳法的应用课件 北师大版选修45
