1 . 材料一:有理数都能表示成
,(
,且
,s与t互质)的形式,进而有理数集可以表示为{
且
,s与t互质}.
材料二:我们知道.当
时,可以用一次多项式近似表达指数函数,即
;为提高精确度.可以用更高次的多项式逼近指数函数.
设
对等式两边求导,
得![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46fff2ffb69cbf301c9efca778fa2636.png)
对比各项系数,可得:
,
,
,…,
;
所以
,取
,有
,
代回原式:
.
材料三:对于公比为
的等比数列
,当
时,数列
的前n项和
.
阅读上述材料,完成以下两个问题:
(1)证明:无限循环小数3.7为有理数;
(2)用反证法证明:e为无理数(e=2.7182^为自然对数底数).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/537c64844b32a708d299ff92dc53c747.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0948ca0227d20b76a27cd1a6d65527fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/823ab696d27d40920c39b8c910789380.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00957293044aadf33411d25f96a33922.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/823ab696d27d40920c39b8c910789380.png)
材料二:我们知道.当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ba156ab181b28fa42e7e4596e69c4d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8f0237baa1472e643b6654cd8efe601.png)
设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/860870ed643c19574d5d8b3a01b6afca.png)
得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46fff2ffb69cbf301c9efca778fa2636.png)
对比各项系数,可得:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a819b1551fee7d49f197b6c7db77a495.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93dfb46889c0485f74277e329d8c5ec8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/233ede8e2b7ddd6807e67d974b7370ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8feaae3fe8a0a3504ce8f2daee1d0a50.png)
所以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51190247f6103b03b31a4f6f01420ddf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c7717db429760899f23de4d22702543.png)
代回原式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b4da092e325d22a89c38348dd5bae89.png)
材料三:对于公比为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6520ff48dba646ba8b7a7d7ae7ca35bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a107eb946e0fe41629c644b7628d5cba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12ecdb961754406f92fceddd4f77cfd3.png)
阅读上述材料,完成以下两个问题:
(1)证明:无限循环小数3.7为有理数;
(2)用反证法证明:e为无理数(e=2.7182^为自然对数底数).
您最近一年使用:0次
2 . 求正整数
,使得
成立.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63022eef355eb0776a1ee956c8fdf8e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab797c5e8378285cff446f8bca5e9012.png)
您最近一年使用:0次
名校
3 . 下列关于排列组合数的等式或说法正确的有( )
A.![]() |
B.设![]() ![]() |
C.已知![]() ![]() ![]() ![]() |
D.等式![]() ![]() |
您最近一年使用:0次
2023-03-28更新
|
1811次组卷
|
5卷引用:江苏省常州市前黄高级中学2023届高三考前攀登行动(一)数学试题
江苏省常州市前黄高级中学2023届高三考前攀登行动(一)数学试题(已下线)第九章 综合测试B(基础卷)(已下线)专题10 计数原理 (分层练)江苏省常州市第一中学2022-2023学年高二下学期3月阶段检测数学试题(已下线)专题01 两个计数原理与排列组合(7类压轴题型)-【常考压轴题】2023-2024学年高二数学压轴题攻略(人教A版2019选择性必修第三册)
解题方法
4 . 函数
满足
,令
,对任意的
,都有
,若
,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e41fb84bb4043552dd0ad25fd9aed0e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7aa0766aba16fa6ebe08c40bfefeb6d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e541ea2f855f981c96207070683d388.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c62aaec3e5e237e6b0a9d9d4d8b6fce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb14af983d72723a397142b672fd3cc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c69303bd9eccb3ccb55c9e4cd03a8a3.png)
A.![]() | B.3 | C.1 | D.![]() |
您最近一年使用:0次
名校
5 . 给定一个n项的实数数列
,任意选取一个实数c,变换
将数列
变换为数列![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d5280b3866264f8c266fbcfc3de96fe.png)
,再将得到的数列继续施行这样的变换,这样的变换可以连续施行多次,并且每次所选择的实数c可以不相同,将第
次变换记为
,其中
为第
次变换时选择的实数.如果通过k次变换后,数列中的各项均为0,则称
为“
次归零变换”,如
项数列
有“
次归零变换”
.
(1)对数列
,请给出其一个“
次归零变换”,其中
;
(2)求证:对任意
项数列,都存在“
次归零变换”;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f53032c9a9a3edca24a7ed909e29a58b.png)
(3)分别判断两个数列
与![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b09a30c1f614670c58260ac05637fa6.png)
是否存在“
次归零变换”,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71440f7f7adb82343de4e1fe714844c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15e5927e6b179a7df32697f18ba04b41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d5280b3866264f8c266fbcfc3de96fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd261890430da46de995859443c31a0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d0c165709cb397edbdf03aecee92bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5925987d5a6eb1ce1fec68b810cb6fcb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9e5dfcc28321b563a8012ec2899c502.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ddb41001678e62a7f251aca18ba87223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/392131fb1129db593075a41e42614c71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1925f666f670a98809ce07b2db292642.png)
(1)对数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63541441d26f277d9ed7816c089c6b6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32122984badd2651714490544ab42682.png)
(2)求证:对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfc321599521a98661ed719cc82ca87c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f53032c9a9a3edca24a7ed909e29a58b.png)
(3)分别判断两个数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0f28c61ca6fda7a77fdae8b39b41519.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b09a30c1f614670c58260ac05637fa6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c4259bd8918d091c52eb0bd5e3e0805.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3493454f81c555d65d46357d719c5fdf.png)
您最近一年使用:0次
名校
6 . 已知
,其中
.
(1)当
时,分别求
和
时
的单调性;
(2)求证:当
时,
有唯一实数解
;
(3)若对任意的
,
都有
恒成立,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dae1d5e1ae68c2c1f7b46f65263c3c71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22dd8b3dc4c609bab82d356a5cc2208d.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c87b351f16728b0023fd63678f8103c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc2d3df37e73a8abea815f37dbb3fff5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)求证:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3047d4ab078dafc06c047bcbf0a6ffaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
(3)若对任意的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ab958eede2dbad749ba70bb230c88fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22e38c541dec8fce1d26886e5ef7d21f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
您最近一年使用:0次
2023-02-04更新
|
730次组卷
|
2卷引用:江苏省常州市华罗庚中学2022届高三下学期3月模拟数学试题
7 . 给正方体的八个顶点涂色,要求同一条棱的两个端点不同色,现有三种颜色可供选择,不同的涂色方法有________ 种.
您最近一年使用:0次
2022-09-29更新
|
2250次组卷
|
6卷引用:专题10-1 排列组合20种模型方法归类-2
(已下线)专题10-1 排列组合20种模型方法归类-2(已下线)考点03 排列组合的综合 2024届高考数学考点总动员【练】(已下线)专题8-1排列组合归类-1浙江省学军中学紫金港校区、海创园校区2021-2022学年高二下学期期中数学试题(已下线)6.2.3 组合(分层作业)-【上好课】2022-2023学年高二数学同步备课系列(人教A版2019选择性必修第三册)(已下线)重难点:排列组合综合检测(提高卷)-【同步题型讲义】2022-2023学年高二数学同步教学题型讲义(人教A版2019选择性必修第三册)
8 . 类比排列数公式
,定义
(其中
,
),将右边展开并用符号
表示
(
,
)的系数,得
,则:
(1)![](https://staticzujuan.xkw.com/quesimg/Upload/formula/510ccf3b1162478d7e998aca3e0c6750.png)
______ ;
(2)若
,
(
,
),则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/314b48a8814396398314e6e3c6239e22.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d54e5eda816e52f8c36d7e1d8827f9bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18343bcfe782eae3086f44f992c83a12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e145b6046bc80d0ffecc61ac67c87ca1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb63478132d4c1fef3c17e591919da83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1674b83ede4a3e742656179310ee50c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ce733aaa0bf65084fedbf529da36ef9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7601dbefa6836756e3d2731b79af0126.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cf5776ec7059c208daf01ca48a34915.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ef5323d67d80208d0a91ae8f98f8670.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/510ccf3b1162478d7e998aca3e0c6750.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd54b98795d8f7b4555ee2a44aa5edf4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1f73c046ff89b81f1453460c2dd2ed2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/142ca1ff5f26dc668c27ae77e7c366b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9275522a5ac41cc65bf9022a7d320eb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/314b48a8814396398314e6e3c6239e22.png)
您最近一年使用:0次
9 . 设n是正整数,我们说集合
的一个排列
具有性质P,是指在
当中至少有一个i,使得
.求证:对于任何n,具有性质P的排列比不具有性质P的排列的个数多.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27b73ddd744f2f715aad49f52da0aea6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/501c3dd561143eae443ca3bb3d5caf53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1da1428d10c6854c6be55b791fe98fe9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/968b1b4abe8c4b4daa25d174520ddc67.png)
您最近一年使用:0次
10 . 正方体六个面上分别标有A、B、C、D、E、F六个字母,现用5种不同的颜色给此正方体六个面染色,要求有公共棱的面不能染同一种颜色,则不同的染色方案有( )种.
A.420 | B.600 | C.720 | D.780 |
您最近一年使用:0次
2021-09-06更新
|
3197次组卷
|
11卷引用:考点01 排列组合-2022年高考数学(理)一轮复习小题多维练(全国通用)
(已下线)考点01 排列组合-2022年高考数学(理)一轮复习小题多维练(全国通用)(已下线)专题10-4 排列组合小题归类(理)-2022年高考数学毕业班二轮热点题型归纳与变式演练(全国通用)(已下线)专题10-1 排列组合20种模型方法归类-2(已下线)【练】 专题一 排列数、组合数的性质应用问题(压轴大全)福建省莆田第一中学2020-2021学年高二下学期期中考试数学试题(已下线)第02讲 排列-【帮课堂】2021-2022学年高二数学同步精品讲义(苏教版2019选择性必修第二册)湖北省荆州市石首市2021-2022学年高二下学期期中数学试题福建省泉州市安溪一中、养正中学、惠安一中、泉州实验中学四校2021-2022学年高二下学期期中联考数学试题(已下线)高二数学下学期期中精选50题(压轴版)2021-2022学年高二数学下学期考试满分全攻略(人教A版2019选修第二册+第三册)浙江省宁波市北仑中学2022-2023学年高二上学期期中数学试题(1班使用)重庆市乌江新高考协作体2023-2024学年高二下学期第一阶段学业质量联合调研抽测(4月)数学试题