名校
1 . 已知数列
满足,
,
,且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84c58e480cfcab1aff130f7f2d139a6b.png)
.记集合
.
(1)若
,求集合
中元素的个数;
(2)①求证:
,
.
②若集合
中存在一个元素是3的倍数,求证:
中所有元素都是3的倍数;
(3)求集合
中元素个数的最大值,及元素个数最大时不同
的个数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ebe3c2a37685ddf5fb089f963169de6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f793fcd885eac1c218c539688a5a773.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84c58e480cfcab1aff130f7f2d139a6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b23fc7915cd368f53f3f1465e05c9c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcfab5a599d30821f0e3e21a1137e161.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06dd3c3b45125d4b484e2894992610f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(2)①求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72620c113a6fe83273803a9ac24baa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11b5269a64a32095e81eab4bbab5782a.png)
②若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(3)求集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
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2 . 数列
满足
,
(
).
(1)计算
,
,猜想数列
的通项公式并证明;
(2)求数列
的前n项和;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ea8d0e50065114b05ef2dc1ea1129cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be815a472dbc3112591a3c311750b1ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
(1)计算
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/109c4b28048ec9e72ba8ecd6311d9f7e.png)
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名校
3 . 平面上
个圆最多把平面分成
个区域,通过归纳推理猜测
的表达式,再利用数学归纳法证明.用数学归纳法证明的过程中,当
时,需证
( ).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63ba21f3d0cfc86d40e2e06446623ce0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/056a3376fabf84be387cf75f3bba6d2e.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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名校
4 . 用数学归纳法证明命题“
,
时,假设
时成立,证明
时也成立,可在左边乘以一个代数式______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f8365233f341451598eb50525a1557a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ddbd01c4c9bd404bcb217ee7f8639fd6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ef7ca2b3e8061384501f668e59696a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63ba21f3d0cfc86d40e2e06446623ce0.png)
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名校
5 . 利用数学归纳法证明不等式
的过程中,由
变到
时,左边增加了( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a360badb860bc02c2cb0428940b608e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f4b264c00c44679d63c24c145a1bcf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63ba21f3d0cfc86d40e2e06446623ce0.png)
A.1项 | B.![]() | C.![]() | D.![]() |
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23-24高二下·全国·课前预习
6 . 数学归纳法的定义
一般地,证明一个与正整数
有关的命题,可按下列步骤进行:
(1)(归纳奠基)证明当________ 时命题成立;
(2)(归纳递推)以“当________ 时命题成立”为条件,推出“当________ 时命题也成立”.
只要完成这两个步骤,就可以断定命题对从
开始的所有正整数
都成立,这种证明方法称为数学归纳法.
一般地,证明一个与正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(1)(归纳奠基)证明当
(2)(归纳递推)以“当
只要完成这两个步骤,就可以断定命题对从
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16530bfffc3b0bb4bda872bf43a3b82f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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23-24高二下·全国·课前预习
7 . 数学归纳法的操作流程
(1)________ 奠基要稳,有些问题中验证的初始值
不一定为1.
(2)正确分析由
到
时式子________ 是应用数学归纳法成功证明问题的保障.
(3)在第二步证明中一定要________ ,这是数学归纳法证明的核心环节,否则这样的证明就不是利用数学归纳法证明.
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16530bfffc3b0bb4bda872bf43a3b82f.png)
(2)正确分析由
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2b66d04abdc608824821dee4c842065.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e8d70d8c5c609c5b55dd2d795be9648.png)
(3)在第二步证明中一定要
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8 . 已知①设函数
的值域是
,对于
中的每个
,若函数
在每一处
都等于它对应的
,这样的函数
叫做函数
的反函数,记作
,我们习惯记自变量为
,因此
可改成
即为原函数的反函数.易知
与
互为反函数,且
.如
的反函数是
可改写成
即为
的反函数,
与
互为反函数.②
是定义在
且取值于
的一个函数,定义![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db6c5d65e0ccd27748fcbc420c6a2e22.png)
,则称
是函数
在
上的
次迭代.例如
,则
.对于一些相对复杂的函数,为求出其
次迭代函数,我们引入如下一种关系:对于给定的函数
和
,若函数
的反函数
存在,且有
,称
与
关于
相似,记作
,其中
称为桥函数,桥函数满足以下性质:
(i)若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e0e8a92152787274ed6e06a21ef1661.png)
(ii)若
为
的一个不动点,即
,则
为
的一个不动点.
(1)若函数
,求
(写出结果即可)
(2)证明:若
,则
.
(3)若函数
,求
(桥函数可选取
),若
,试选取恰当桥函数,计算
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e66c24e657d998beb013ad1fb311d33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f93e3581e920716e710e22b31006bdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f93e3581e920716e710e22b31006bdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63ced31d098cfb0cf14d906e97e6353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e66c24e657d998beb013ad1fb311d33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/955f3c2b80eebc3f88c804112e5f41f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/955f3c2b80eebc3f88c804112e5f41f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb87ce86e2c9a1a8188b03b74438fdd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb87ce86e2c9a1a8188b03b74438fdd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e66c24e657d998beb013ad1fb311d33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25ef38135b0e7906687d8a4918a4cb67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99b161347f6a2fcfd9bf0acf1e8a03fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43737e3ca063dfc210d0c72924a4930.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/808bd2fc4b344e7669fca65b4fa122df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99b161347f6a2fcfd9bf0acf1e8a03fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/808bd2fc4b344e7669fca65b4fa122df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99b161347f6a2fcfd9bf0acf1e8a03fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db6c5d65e0ccd27748fcbc420c6a2e22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c110a1293773729278a214c7fe8d544e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a33cfe27fd2276a7c542f062c17b4d85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/052ddf3664af9ab2990f3ea622997e00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/910ca4e4f009554b599eab90e1d94c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/544f91d4fb22c571db9f8481b72a0419.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71567deb76e48f8a2424b06536cbe465.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a66b033ee7a03c7b3508583481465275.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/544f91d4fb22c571db9f8481b72a0419.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1249f186df944244da02e1b8c754005.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/544f91d4fb22c571db9f8481b72a0419.png)
(i)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1249f186df944244da02e1b8c754005.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e0e8a92152787274ed6e06a21ef1661.png)
(ii)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66f66a2b3d90f0d935d6c8ebaf675349.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4f199ad3fad8657afa38f370b319a75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
(1)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/192bb45bd15b200f40b34377bc58905b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a33cfe27fd2276a7c542f062c17b4d85.png)
(2)证明:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1249f186df944244da02e1b8c754005.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99315f5b2ae9bea18e06401b41d3780c.png)
(3)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/becd598a11b876d858728161a7a09705.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a33cfe27fd2276a7c542f062c17b4d85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04ab7717944da2b6cc305b6a65f91408.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c85f9d9be0ba965ff7beb0e011267f29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bee7f1ccd52c7d526b6d466b970e769.png)
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9 . 用数学归纳法证明“
”的过程中,从
到
时,左边增加的项数为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdc0dfae24a9d5e405673a131b120927.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4144cc65da072c3f9e149c1d524369a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63ba21f3d0cfc86d40e2e06446623ce0.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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10 . 相传古希腊毕达哥拉斯学派的数学家常用小石子在沙滩上摆成各种形状来研究数,并根据小石子所排列的形状把数分成许多类.现有三角形数表按如图的方式构成,其中项数
,第一行是以1为首项,2为公差的等差数列.从第二行起,每一个数是其肩上两个数的和,例如:
;
为数表中第
行的第
个数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4f4c5a9887ac923aaab6dd942cf0273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2032083f2e82474fc2ec2d755459a64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a935cfef7ed524cf2ff73fd661e1ea9c.png)
……
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e9a121e0c62dd80c771e0bb057771d4.png)
(1)求第2行和第3行的通项公式
和
;
(2)一般地,证明一个与正整数
有关的命题,可按下列步骤进行:①证明当
时命题成立;②以“当
时命题成立”为条件,推出“当
时命题也成立.”完成这两个步骤就可以断定命题对
开始的所有正整数
都成立,这种方法即数学归纳法.请证明:数表中除最后2行外每一行的数都依次成等差数列,并求
关于
的表达式;
(3)若
,
,试求一个等比数列
,使得
,且对于任意的
,均存在实数
,当
时,都有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5d0a73f50b3e4583f1c1b6d6bf0d18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6831b015f2f16c3439bfca2a9ecea6ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2a57936aa3c10e1045536f9c2ad37e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7600d2cfbdc6146db96cc545706004f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4f4c5a9887ac923aaab6dd942cf0273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2032083f2e82474fc2ec2d755459a64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a935cfef7ed524cf2ff73fd661e1ea9c.png)
……
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e9a121e0c62dd80c771e0bb057771d4.png)
(1)求第2行和第3行的通项公式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aea009aa1b893f59585cc2ec5dfede2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff7e065e93a47524854d9e3e50876b10.png)
(2)一般地,证明一个与正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5bb1f8d351dd6d2f27064908a5f00a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16743b46792d3250ede27f695612003a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63ba21f3d0cfc86d40e2e06446623ce0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d7e9f86738335a22298559db41037a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec9d1cd31d3fa069693c285262739a43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27e07c547da901b07c141cddbe0013fb.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50218cf491febde222900c18de34037b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4a92d4463e0a56109a13d60b640e0a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04d63b4673a90a76adf4171e09d0382e.png)
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