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3 3.2 2用数学归纳法证明不等式用数学归纳法证明不等式,贝努利不贝努利不等式等式目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航1.会用数学归纳法证明简单的不等式.2.会用数学归纳法证明贝努利不等式.3.了解贝努利不等式的应用条件.目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航1.用数学归纳法证明不等式在不等关系的证明中,有多种多样的方法,其中数学归纳法是最常用的方法之一,在运用数学归纳法证不等式时,推导“k+1”成立时,比较法、分析法、综合法、放缩法等方法常被灵活地应用.【做一做1-1】 欲用数学归纳法证明:对于足够大的正整数n,总有2nn3,n0为验证的第一个值,则()A.n0=1B.n0为大于1小于10的某个整数C.n010D.n0=2解析:n=1时,21;n=2时,48;n=3时,827;n=4时,1664;n=5时,32125;n=6时,64216;n=7时,128343;n=8时,256512;n=9时,5121 000.故选C.答案:C目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航【做一做1-2】 用数学归纳法证明“ nN*,n1)”时,由n=k(k1)不等式成立推证n=k+1时,左边应增加的项数是()A.2k-1B.2k-1C.2kD.2k+1解析:增加的项数为(2k+1-1)-(2k-1)=2k+1-2k=2k.答案:C目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航2.用数学归纳法证明贝努利不等式(1)定理1(贝努利不等式):设x-1,且x0,n为大于1的自然数,则(1+x)n1+nx.(2)定理2:设为有理数,x-1,若01,则(1+x)1+x;若1,则(1+x)1+x.当且仅当x=0时等号成立.名师点拨当指数推广到任意实数且x-1时,若01,则(1+x)1+x;若1,则(1+x)1+x.当且仅当x=0时等号成立.目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航应用数学归纳法证明不等式,从“n=k”到“n=k+1”证明不等式成立的技巧有哪些?剖析: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SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航题型一题型二题型三用数学归纳法证明数列型不等式 (1)求数列an的通项公式;(2)求证:对一切正整数n,不等式a1a2an12=1;当n=2时,22=4=22;当n=3时,23=852=25;当n=6时,26=6462=36.故猜测当n5(nN*)时,2nn2.下面用数学归纳法进行证明:(1)当n=5时,显然成立.(2)假设当n=k(k5,且kN*)时,不等式成立,即2kk2(k5),则当n=k+1时,2k+1=22k2k2=k2+k2+2k+1-2k-1=(k+1)2+(k-1)2-2(k+1)2(因为(k-1)22).目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航题型一题型二题型三反思利用数学归纳法比较大小,关键是先用不完全归纳法归纳出两个量的大小关系,猜测出证明方向,再利用数学归纳法证明结论成立.目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航题型一题型二题型三用数学归纳法证明探索型不等式 目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航题型一题型二题型三(1)当n=1时,显然成立.(2)假设当n=k(kN*,且k1)时,目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航题型一题型二题型三反思用数学归纳法解决探索型不等式的思路是:观察归纳猜想证明,即先通过观察部分项的特点进行归纳,判断并猜测出一般结论,然后用数学归纳法进行证明.目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航1 2 3 41下列选项中,不满足12+23+34+n(n+1)3n2-3n+2的自然数n是()A.1B.1,2C.1,2,3 D.1,2,3,4解析:将n=1,2,3,4分别代入验证即可.答案:C目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航1 2 3 4答案:C 目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航1 2 3 4目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理DIANLITOUXI典例透析SUITANGLIANXI随堂演练ZHONGNANJUJIAO重难聚焦ZHISHISHULI知识梳理目标导航1 2 3 4
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