1 . 对于项数为
的数列
,若数列
满足
,
,其中,
表示数集
中最大的数,则称数列
是
的
数列.
(1)若各项均为正整数的数列
的
数列是
,写出所有的数列
;
(2)证明:若数列
中存在
使得![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cce3b6654490dcd8177970631e929d3d.png)
,则存在
使得
成立;
(3)数列
是
的
数列,数列
是
的
数列,定义
其中
.求证:
为单调递增数列的充要条件是
为单调递增数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb951647594b67c82045b7ba69f57cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c3bd239656c2ef509ed9f2a91a68317.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfd88e35dc6c2b82b4bb29475d37c015.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cda05b106f920bbcfa02320229ca3c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/698f45c9ed5bb04924f1037107e76988.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c3bd239656c2ef509ed9f2a91a68317.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb951647594b67c82045b7ba69f57cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(1)若各项均为正整数的数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb951647594b67c82045b7ba69f57cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c2f32a17f0261d32079efed31d414a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb951647594b67c82045b7ba69f57cd.png)
(2)证明:若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb951647594b67c82045b7ba69f57cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50a272adba0f1120109824440f0e252c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cce3b6654490dcd8177970631e929d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6afadbd5dc4d8003ac2a0c85678dbecf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4578089af6806bf1257491091b924d3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c2dd6a827492dffddd07e621a4bbe36.png)
(3)数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c3bd239656c2ef509ed9f2a91a68317.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb951647594b67c82045b7ba69f57cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4af75cc7d8cc976dce8bf9bd8fdc18f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/defa9f328d5108bbec5c56eebfe95567.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d019d9549df6aed0dab378301d889ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70322214e5c9d9a8df10eb45930f5745.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cb2e86a5aa896bef041701e0e1771ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8504784fec514a92d845910c6721c3a.png)
您最近一年使用:0次
22-23高二下·上海·期末
2 . 对于任意的
,若数列
同时满足下列两个条件,则称数列
具有“性质m”:
①
;②存在实数M,使得
成立.
(1)数列
、
中,
,判断
、
是否具有“性质m”;
(2)设各项为正数的等比数列
的前n项和为
,且
,
,数列
是否具有“性质m”,若具有,请证明你的猜想,并指出M的范围;若不具有,理由?
(3)若数列
的通项公式
.对于任意的
(
),数列
具有“性质m”,且对满足条件的M的最小值
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15a70b95c53fb6655721e2a8c61f5c2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc75a9da38151496ca2adce84a977b96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5612ce06759d0f77ca029d10083f7d1e.png)
(1)数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef91b91fc7ac183837bd7c6799f19c64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)设各项为正数的等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12baad2468950ea89781e38432d88f01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22ab4706be6b3854b9c30ab609e5da68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/846fa57d92d6ad44d6a0cafad1e71ed4.png)
(3)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9050ccff2f6fea858342b0fb290ad7bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15a70b95c53fb6655721e2a8c61f5c2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8191d51029a192d504bf1b736029f82e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f66884efff7400f92b530d69d029778d.png)
您最近一年使用:0次
2023高三·全国·专题练习
3 . 已知函数
的图象按向量
平移后得到
的图象,数列
满足
(
且
).
(1)若
,满足
,求证:数列
是等差数列;
(2)若
,试判断数列
中是否存在最大项与最小项,若存在,求出最大项与最小项,若不存在,请说明理由;
(3)若
,试证明:数列
单调递减,且
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eda448839c7a87c83aa4fe117b4cae69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffe34cc27c9686bf65697ebd8bb6dadc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbc5ac1542da56851d19240e723a1530.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65a40142c84be68ee2918c3a8303388c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc6545b8eca1c4223ed701a199a85683.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88d48868b259993d0000b7c47525ebcb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc6545b8eca1c4223ed701a199a85683.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95cae75fa078f0961c2966220d895b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ebd37e349df476bbf58a23fa8be26bf.png)
您最近一年使用:0次
22-23高二上·上海·期中
解题方法
4 . 已知函数
的图象按向量
平移后得到
的图象,数列
满足
(
且
).
(1)若
,满足
,求证:数列
是等差数列;
(2)若
,试判断数列
中是否存在最大项与最小项,若存在,求出最大项与最小项,若不存在,请说明理由;
(3)若
,试证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eda448839c7a87c83aa4fe117b4cae69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffe34cc27c9686bf65697ebd8bb6dadc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbc5ac1542da56851d19240e723a1530.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65a40142c84be68ee2918c3a8303388c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc6545b8eca1c4223ed701a199a85683.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88d48868b259993d0000b7c47525ebcb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc6545b8eca1c4223ed701a199a85683.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95cae75fa078f0961c2966220d895b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ebd37e349df476bbf58a23fa8be26bf.png)
您最近一年使用:0次
2022-11-16更新
|
758次组卷
|
4卷引用:专题04 数列(10个考点)【知识梳理+解题方法+专题过关】-2022-2023学年高二数学上学期期中期末考点大串讲(沪教版2020必修第三册+选修一)
(已下线)专题04 数列(10个考点)【知识梳理+解题方法+专题过关】-2022-2023学年高二数学上学期期中期末考点大串讲(沪教版2020必修第三册+选修一)(已下线)高二下期中真题精选(压轴40题专练)-【满分全攻略】2022-2023学年高二数学下学期核心考点+重难点讲练与测试(沪教版2020选修一+选修二)(已下线)期中真题必刷压轴50题专练-【满分全攻略】2023-2024学年高二数学同步讲义全优学案(沪教版2020必修第三册)(已下线)专题1 数学归纳法及其变种 微点3 数学归纳法综合训练
5 . 设集合
由满足下列两个条件的数列
构成:①
;②存在实数
,使
为正整数)
(Ⅰ)在只有5项的有限数列
、
中,其中
,
,
,
,
,
,
,
,
,
,试判断数列
、
是否为集合
中的元素;
(Ⅱ)设
是等差数列,
是其前
项和,
,
,证明数列
,并写出
的取值范围;
(Ⅲ)设数列
,对于满足条件的
的最小值
,都有
求证:数列
单调递增.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91edc7e2d4811f5ea6c01284cf00393a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc75a9da38151496ca2adce84a977b96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de938b541709fad66555cbda07bb818e.png)
(Ⅰ)在只有5项的有限数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fce83115a50f99e08e9a2db7267aeed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c9b6e51986fe5d7a7265e0e93adcb4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ced4e381e8c3336848b8c436dbc584f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0748c346ed88f98e424de8edf278325.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36ba808c24aeae6a2f34b98ae5ec04ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59dd6c97d2ee3e74ba5730f1cbcc1d43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eab7d59ce066c8f0b346719003f8e28f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77d9bd40057948c5e3eb23064a673284.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd4cf909f2febeea7d169459d0cf0bcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f409a81dd85e0f1f845ed2ec77cf040c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fce83115a50f99e08e9a2db7267aeed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91edc7e2d4811f5ea6c01284cf00393a.png)
(Ⅱ)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38ef4c4439b36c2847b0056a116d56d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3903653955c424d3f6135edc5b47e231.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d6fc0f7bf298786fcb97a8906ccea26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3887ca2c727a713f179fe48bcfcc742e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52dcd2fa7adff65e3864f2d42370e6c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(Ⅲ)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32d47dc0629b2277ed4b571e1a9a9880.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fb44db1dc864ff4901be1e10da79747.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c576e76cfa2bacb5a303fd6a5e053b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d813f3ca8db41a4db6c18eac30fef98.png)
您最近一年使用:0次
20-21高二上·全国·单元测试
解题方法
6 . 设集合W由满足下列两个条件的数列{an}构成:①
;②存在实数M,使an≤M(n为正整数)
(1)在只有5项的有限数列{an}、{bn}中,其中a1=1,a2=2,a3=3,a4=4,a5=5,b1=1,b2=4,b3=5,b4=4,b5=1,试判断数列{an}、{bn}是否为集合W中的元素;
(2)设{cn}是等差数列,sn是其前n项和,c3=4,s3=18,证明数列{sn}∈W,并写出M的取值范围;
(3)设数列{dn}∈W,对于满足条件的M的最小值M0,都有dn≠M0(n∈N*)求证:数列{dn}单调递增.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc75a9da38151496ca2adce84a977b96.png)
(1)在只有5项的有限数列{an}、{bn}中,其中a1=1,a2=2,a3=3,a4=4,a5=5,b1=1,b2=4,b3=5,b4=4,b5=1,试判断数列{an}、{bn}是否为集合W中的元素;
(2)设{cn}是等差数列,sn是其前n项和,c3=4,s3=18,证明数列{sn}∈W,并写出M的取值范围;
(3)设数列{dn}∈W,对于满足条件的M的最小值M0,都有dn≠M0(n∈N*)求证:数列{dn}单调递增.
您最近一年使用:0次
7 . 设向量
,
(
为正整数),函数
在
上的最小值与最大值的和为
.又数列
满足:
.
(1)求证:
;
(2)求
的表达式;
(3)若
,试问数列
中,是否存在正整数
,使得对于任意的正整数
,都有
成立?证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c4f40def777e1d7c7ced828899ed593.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fecd0618a94b609ba39648aee5078ad7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3001a6338718284f7d39dc9a0c9b1ccd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e11f4ca0e7ace69f92130d0525bcdb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f422217788237b19729bece6bbc07687.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc193f718a5f5fa18880eedfe45b24d2.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4957e76e086cebb2ceb534b6a3dd907.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7f69e2573d086fa7cfbd7e4f5a162c1.png)
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8 . 对于正整数m,n,存在唯一的自然数a,b,使得
,其中
,我们记
.对任意正整数
,定义
的生成数列为
,其中![](https://staticzujuan.xkw.com/quesimg/Upload/formula/874689ac8a6f70ecd93aa1aaf7d626f6.png)
.
(1)求
和
.
(2)求
的前3项.
(3)存在
,使得
,且对任意
成立.考虑
的值:当
时,定义数列
的变换数列
的通项公式为
当
时,定义数列
的变换数列
的通项公式为
若数列
和数列
相同,则定义函数
,其中函数
的定义域为正整数集.
(ⅰ)求证:函数
是增函数.
(ⅱ)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bad12b60594a6408ccc237cad2880088.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ff38a5b95f1fb452d87fce9c80b1249.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/574aa06d3d4f9952157c60ec8f40b839.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec5914cd027a87542e12cfb0c92c0aa1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/874689ac8a6f70ecd93aa1aaf7d626f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57dbb663375b927f3c00d1be3fda1508.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37dbd8745cf6246eea337a1bf91258c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c87dc8de41005802093ae66c08f46a7a.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/926145a2b2232a88ceed6e69dc050265.png)
(3)存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d7e9f86738335a22298559db41037a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/623ebc0f0eba98e8f63b7cd982c11009.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8144524289a76dadf88c02976acea2db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebf0ce448d146abfcd0bd689b925d168.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c20d94765a05ee86edf370dd64fcff7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec5914cd027a87542e12cfb0c92c0aa1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d054e320d843bf0ca4a620519f75ebaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8121a889da1f3ecbecd5387a9473c0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d98b98330b62847c220ab2f127e11391.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec5914cd027a87542e12cfb0c92c0aa1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d054e320d843bf0ca4a620519f75ebaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/caf19212dd24556423636ce37034cdf7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d054e320d843bf0ca4a620519f75ebaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96dbc1106da9238ec6300e13ca3d9b10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46498efb2e485c1172abe2189e498613.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b88ba8e1dc91318d8a9e1856d8ae080e.png)
(ⅰ)求证:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b88ba8e1dc91318d8a9e1856d8ae080e.png)
(ⅱ)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bee5e2ada7f1e99bf68508d21b8f6fb7.png)
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9 . 数列
中,
.
(1)求证:数列
是等差数列;
(2)若
,都有
恒成立,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a918011b01d617a8cc90cc353c84bc9d.png)
(1)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eed5d4cb13b25787a125847474afae87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a63abf79ae4f45753069484ab7daf71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
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10 . 基本不等式:对于2个正数
,它们的算术平均数不小于它们的几何平均数,即
,当且仅当
时,等号成立.可以推广到一般的情形:对于
个正数
,它们的算术平均数不小于它们的几何平均数,
.当且仅当
时,等号成立.若无穷正项数列
同时满足下列两个性质:①
;②
为单调数列,则称数列
具有性质
.
(1)若
;求数列
的最小项;
(2)若数列
的前
项和为
,判断数列
是否具有性质
,并说明理由;
(3)若
,求证:数列
具有性质
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0158862238e250d2a2598b7d4ecd148.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/140564fb2e11f8411e353d2fa73fbee9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58b2fb1017d5dfe3a300de3e14a71b6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efff8ec14cb242e793afab4468bf2e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2617515e5ce81b3f5d9f4e806b21b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6879960be91ea52297d587e9a014f54a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17fac66473a039bdb47c2a248b0f4854.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b9bb1963d176a00c82828c93ca0e2c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d013861990cf331c82eb453416ae31bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4247739746b8ddf1403541047e8b5580.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5ab0309e2cd35585ea9fb2cc3017abf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
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