2023高三·全国·专题练习
1 . 已知m,n为正整数.
(1)用数学归纳法证明:当
时,
;
(2)对于
,已知
,求证:
;
(3)求出满足等式
的所有正整数n.
(1)用数学归纳法证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adc3e5be1796493161a4df7e28a6f6b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4b6e1ca4dd76952e3544b14978bd359.png)
(2)对于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/831608f09609c37f757f5bfcd01253f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/186c794ebbde3237056af29cb97778f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91cbed0668c7d57785e9d95b5775e41f.png)
(3)求出满足等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91192b519c2eb79c20d0205bc22016a2.png)
您最近一年使用:0次
名校
2 . 定理(三角不等式),对于任意的
、
,恒有
.定义:已知
且
,对于有序数组
、
、
、
,称
为有序数组
、
、
、
的波动距离,记作
,即
,请根据上述俼息解决以下几个问题:
(1)求函数
的最小值,并指出函数取到最小值时
的取值范围;
(2)①求有序数组
、
、
、
的波动距离
;
②求证:若
、
、
、
且
,则
;题(注:该命题无需证明,需要时可直接使用).设两两不相等的四个实数
、
、
、
,求有序数组
、
、
、
的波动距离
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dd0914dc4d4c7f75710ff460a286fcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49506c61cf5c61605f1cf90a440348cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd4613271f782a90ab580131d09d03d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dec475a4298eab592d6589aab8915276.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ef141315bf951ddcd300f0743a16897.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7cfd590897d8d908066c781c63a812d.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d3b887215cd1514d3e2e79063729a4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(2)①求有序数组
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d91e07104b699c4012be2d26160976a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7330e52932883877de428cfe91962b96.png)
②求证:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a876ecb804eb0553c246e5fcc40b708.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b73abfe4bc26b1ded680d7abb1a2cac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/effb89a4bffb74028211ecfe671b79d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](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/d46944e1594eec140cacd7b454342561.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](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/daf464629fa321a6ff7401ab79f07083.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cc0ce632fa217dc77f6c92afd311815.png)
您最近一年使用:0次
2022-08-22更新
|
416次组卷
|
7卷引用:专题02 等式与不等式(练习)-2
(已下线)专题02 等式与不等式(练习)-2上海市控江中学2021-2022学年高一上学期期中数学试题上海市高桥中学2022-2023学年高一上学期期中数学试题(已下线)期中模拟预测卷03(测试范围:前三章)-2022-2023学年高一数学上学期期中期末考点大串讲(沪教版2020必修第一册)(已下线)上海高一上学期期中【压轴42题专练】(2)(已下线)第二章 等式与不等式(压轴题专练)-速记·巧练(沪教版2020必修第一册)上海市吴淞中学2023-2024学年高一上学期期中数学试题
名校
3 . 证明:
(1)若
,则
;
(2)求证:当
为正数时,
.
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7ad6cf9fb725f7947bddaf3149ba07d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ca34726fb86e4bdeeb4e87778148bf3.png)
(2)求证:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43545d069f7d7c5f87bfdcad4ce63801.png)
您最近一年使用:0次
2022-04-08更新
|
362次组卷
|
2卷引用:吉林省实验中学2021-2022学年高三下学期最后一次模拟考试数学(理)试题
名校
解题方法
4 . 选用恰当的证明方法,证明下列不等式.
(1)已知实数
,
均为正数,求证:
.
(2)已知
,
都是正数,并且
,求证:
.
(1)已知实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5069dfceae48573f4991a1fa2f45b5c7.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2958030ec9d7543dda1f529593a915e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e44a6abb0974fa7a3ff0477c4e891e0.png)
您最近一年使用:0次
2021-02-06更新
|
545次组卷
|
2卷引用:西藏昌都市第一高级中学2022届高三下学期入学考试数学试题
名校
5 . 已知数列{an}满足a1=2,
(n∈N*).
(1)求证:数列
是等比数列;
(2)比较
与
的大小,并用数学归纳法证明;
(3)设
,数列{bn}的前n项和为Tn,若Tn<m对任意n∈N*恒成立,求实数m的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a04e45f0f7233e1766ba93f36fafb0f3.png)
(1)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0210bf1fb13af42d057c1cf7ccdf7e92.png)
(2)比较
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/245460a7f2be54fa45095316e71014a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0763ff5f577b56744a5969dd1ab8f86.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe115795f19a35c719a10c729edd9885.png)
您最近一年使用:0次
2020-10-27更新
|
822次组卷
|
11卷引用:专题6.6 数学归纳法(讲)- 浙江版《2020年高考一轮复习讲练测》
(已下线)专题6.6 数学归纳法(讲)- 浙江版《2020年高考一轮复习讲练测》(已下线)专题7.6 数学归纳法(讲)-2021年新高考数学一轮复习讲练测(已下线)专题08 数列的通项、求和及综合应用 第一篇 热点、难点突破篇(练)-2021年高考数学二轮复习讲练测(浙江专用))(已下线)专题28 证明不等式的常见技巧-学会解题之高三数学万能解题模板【2022版】【校级联考】浙江省嘉兴市第一中学、湖州中学2018-2019学年高一下学期期中考试数学试题【校级联考】浙江省浙北G2期中联考2018学年高一第二学期数学试题浙江省浙北G2联考2018-2019学年高一第二学期期中考试数学试题(已下线)第四章++数列2(能力提升)-2020-2021学年高二数学单元测试定心卷(人教A版2019选择性必修第二册)(已下线)第四章++数列1(能力提升)-2020-2021学年高二数学单元测试定心卷(人教A版2019选择性必修第二册)(已下线)第04讲 数学归纳法-【帮课堂】2021-2022学年高二数学同步精品讲义(苏教版2019选择性必修第一册)(已下线)第04讲 数学归纳法(核心考点讲与练)-2021-2022学年高二数学考试满分全攻略(人教A版2019选修第二册+第三册)
6 . (1)已知
是实数,求证:
.
(2)用分析法证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0fffbec1fe851795dfdd448bf0d165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69d21cedbf84856328405de3bf6275ba.png)
(2)用分析法证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14c19d94ff48082c1cd213c82c99abf0.png)
您最近一年使用:0次
2020-08-04更新
|
107次组卷
|
10卷引用:专题23 不等式选讲-2020年高考数学(理)母题题源解密(全国Ⅲ专版)
(已下线)专题23 不等式选讲-2020年高考数学(理)母题题源解密(全国Ⅲ专版)(已下线)专题23 不等式选讲-2020年高考数学(文)母题题源解密(全国Ⅲ专版)江西省山江湖协作体2019-2020学年高二上学期第三次月考(统招班)数学(文)试题安徽省蚌埠二中2019-2020学年高二下学期开学检测文科数学试题贵州省兴仁市凤凰中学2019-2020学年高二下学期第一次月考数学(理)试题贵州省兴仁市凤凰中学2019-2020学年高二下学期第一次月考数学(文)试题安徽省淮南一中2020-2021学年高二下学期第一次段考理科数学试题(已下线)2.2.1 直接证明-2020-2021学年高二数学(理)课时同步练(人教A版选修2-2)山西省怀仁市大地学校2020-2021学年高二下学期第三次月考数学(理)试题山西省怀仁市大地学校2020-2021学年高二下学期第三次月考数学(文)试题
解题方法
7 . 证明:(1)已知a,b,
,
,求证:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/135125d796a469155fc4a22dc6be3d10.png)
(2)已知a,b,
,
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ede693e9fed26c40f6fee9c3aaad147c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/751e274e9107d780c39ba9c49d6daefb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/135125d796a469155fc4a22dc6be3d10.png)
(2)已知a,b,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ede693e9fed26c40f6fee9c3aaad147c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/751e274e9107d780c39ba9c49d6daefb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17f917a19a15bceb9a3769e59e25dd9c.png)
您最近一年使用:0次
2020-09-01更新
|
207次组卷
|
2卷引用:山西省运城市高中联合体2022届高三下学期第四次模拟数学(文)试题
8 . 已知数列
满足:
,
.
(1)求证:
时,
;
(2)记
,
,求证:
;
(3)在(2)的条件下,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4791a3aa5bc76e309bc46efa1e15a52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f91f6478219ac36d487e8bb204b1f908.png)
(2)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/457f75159c7d1a6489ffcf3c2d2eda17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e7dcca2acb8fb6e6a6933a02e0a130b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc1f5d407c0e99344ed5f0f5926c5d22.png)
(3)在(2)的条件下,证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00fb75ddd3accc66959494f44d1aebcc.png)
您最近一年使用:0次
名校
9 . 若函数
满足:对任意实数
以及定义中任意两数
、
(
),恒有
,则称
是下凸函数.
(1)证明:函数
是下凸函数;
(2)判断
是不是下凸函数,并说明理由;
(3)若
是定义在
上的下凸函数,常数
,满足:
,
,且
,求证:
,并求
在
上的解析式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4799629218b4b62ffa4082b96888e3c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33bd24e647a626899a243a3f3984f90a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1538a4b84a99b2da4de9600fc5552c50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(1)证明:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/681d6d27b23b1c41834d7516122f73f9.png)
(2)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f45afdf4d717bb03adac6b899c367acb.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/304226ca50149b49702928e44d565964.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2da2ab1f8b5d3281efb94b763fa74081.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9db2cae6cc39553ca2b984741630917.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ceb4645bb34156bfc57de16ec11300f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e61c9a7ed0961f8977a21dab37aab396.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d87cd4403487962c38c8707ba3ab3fa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/304226ca50149b49702928e44d565964.png)
您最近一年使用:0次
10 . 已知
,设多项式
,满足
,
.
(1)求
,
的值;
(2)试探究对于一切正整数
,
是否一定是整数?并证明你的结论;
(3)求证:当
时,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b71ec5f6451593187c2eb9e287bb5fb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b7d0ce400cea7fa51680a320737cd35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/249a976e88133f3b3733f09137cf5c42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffce8c4ae8efb7437586487a8d715884.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
(2)试探究对于一切正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38fcec7af3520884b173b29bda6c657a.png)
(3)求证:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a65e98b404e9f7cf6a39d114526638b4.png)
您最近一年使用:0次