名校
1 . 设非常数数列
满足
,
,其中常数
,
均为非零实数,且
.
(1)证明:数列
为等差数列的充要条件是
;
(2)已知
,
,
,
,求证:数列
与数列
中没有相同数值的项.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d716659722cbc0132626ceab9b404e0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a3cd71690942ef82b8dc04580efc93a.png)
(1)证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcebe948fb198d4fde0df1a1abe680bc.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0733e8dfacbad67bdb7c26930acddaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/234dd79e0081ba0ebd0f7cd4d7d5bef3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad6d8a8a57db1c2fc7f465d2cfd2aa78.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a81e4c91a371984fd3d13330c902b07b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18bc279fef6843dddded8abfa0fbe63e.png)
您最近一年使用:0次
2021-06-08更新
|
791次组卷
|
6卷引用:第17题 数列解答题的两大主题:通项与求和-2021年高考数学真题逐题揭秘与以例及类(新高考全国Ⅰ卷)
(已下线)第17题 数列解答题的两大主题:通项与求和-2021年高考数学真题逐题揭秘与以例及类(新高考全国Ⅰ卷)(已下线)专题08 数列-备战2022年高考数学(文)母题题源解密(全国乙卷)(已下线)查补易混易错点04 数列-【查漏补缺】2022年高考数学三轮冲刺过关(新高考专用)江苏省苏州市吴江区震泽中学2022-2023学年高二10月月考数学试题江苏省南京师范大学《数学之友》2021届高三下学期二模数学试题(已下线)卷09 高二上学期12月阶段测-【重难点突破】2021-2022学年高二数学上册常考题专练(人教A版2019选择性必修第一册)
解题方法
2 . 已知数列
的前
项和为
,
,其中
为常数.
(1)求证:
;
(2)是否存在实数
使得数列
为等比数列,若存在,求出
的值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2627eb8b179fbf5703bc50b60ce29d71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2646e3f088c818898a43682db3a5ffa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd2011d0d84cdd36536bc6c75a714737.png)
(2)是否存在实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
您最近一年使用:0次
名校
3 . 数列
满足:
,
;
(1)求证:
;
(2)求证:对任意正数
,都存在正整数
使得
成立;
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ea8251b5301a3bb8a72a0b5ee408dac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77c8f28af6b58f5d1d356ccdca4674c4.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ad0269b21b7b4cbb8b51345ad3b40fc.png)
(2)求证:对任意正数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef48c5f69c0db267ed83a0df6c2745fc.png)
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6190b53b3c2362929b165929427a535c.png)
您最近一年使用:0次
2022-11-26更新
|
791次组卷
|
6卷引用:上海师范大学附属中学2022-2023学年高二上学期期中数学试题
上海师范大学附属中学2022-2023学年高二上学期期中数学试题(已下线)专题15 数列不等式的证明 微点6 数列不等式的证明综合训练(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)模块一专题3 数列的实际应用和综合问题单元检测篇B提升卷(高二人教B版)(已下线)模块一 专题4 数列的实际应用和综合问题单元检测篇B提升卷(高二北师大版)
名校
解题方法
4 . 已知
为实数,数列
满足:①
;②
.
(1)当
时,求
的值;
(2)求证:存在正整数
,使得
;
(3)设
是数列
的前
项和,求
的取值范围,使数列
为周期数列且方程
有解(若数列
满足:存在
且
,对任意
且
,成立
,则称数列
为以
为周期的周期数列).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7fab51121848ce166035ceab6f4e00b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19e6dcf203a14b46aad3828feb365685.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e65397f11ea8af736f38debadf420c4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3eead98a7980470f3345ccaa8384b9b.png)
(2)求证:存在正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d7e9f86738335a22298559db41037a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3520a1d6ebd326f0484b43ecceb603ed.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86be7eb3aea5635780311d4a98c39f7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb6374313ac48b8425a6ef85dd44441e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/094f977194228bed828f3507f5898934.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ae7d2a51eb86ca377a28decbcb978dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de8610232c77741a37463feba1a66c94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b2b94cbf8f1acc77ed2618d9ba5756a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
您最近一年使用:0次
名校
解题方法
5 . 数列
:
,
,…,
满足:
,
,
或1(
,2,…,
),对任意i,j,都存在s,t,使得
,其中
且两两不相等.
(1)若
,直接写出下列三个数列中所有符合题目条件的数列的序号:
①1,1,1,2,2,2;②1,1,1,1,2,2,2,2;③1,1,1,1,1,2,2,2,2
(2)记
,若
,证明:
;
(3)若
,求n的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.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/d8dc3192c861a4cc44da88f656ae7aa9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/564e60383b05d2e0ee94a733742ae424.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/631c6879b8799ed0f1aefbf28bf988f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5095a28bb1b91bf6bed9e2cfbd76bb18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aadf9ab510510120699c5eee39ab18b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86c4d0383577207858e39b4b19b0853e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/631c70b687b22d032d1cc5050cfc07dc.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e94f16d5ed858699bfea5039a7bf8ae6.png)
①1,1,1,2,2,2;②1,1,1,1,2,2,2,2;③1,1,1,1,1,2,2,2,2
(2)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cb1eff85b93cd753c2a3a4fb9603221.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8a3cc8c48bf54ec8252e5dce6867754.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/743b4f6fde34464397b010cb45eabb7d.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3afa6e51b3b27c3edb330cd7f190b6cf.png)
您最近一年使用:0次
2023-08-05更新
|
739次组卷
|
5卷引用:北京市海淀区清华大学附属中学2022-2023学年高二上学期期中考试数学试题
6 . 已知数列
满足
,
,
.
(1)证明:
是等差数列;
(2)记数列
的前
项和为
,求最小的正整数
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13a6e1d671215fc96e4bee3541d1096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/329785900390130a04a57d0b55aaa569.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ff10a23417557781752064c24437449.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6631307e8ff61b215f447f2527c36e04.png)
(2)记数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41cf1da18d91f7c98086553d157d1a87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f25fad27e74b64d68e6a939bfc2b54d8.png)
您最近一年使用:0次
2022高三·全国·专题练习
解题方法
7 . 若数列
的各项均为正数,对任意n∈N*,
,
为常数,且
.
(1)求
的值;
(2)求证:数列
为等差数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/774f56cbc19ef75c03c3a723c1a9e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4defe7511f052c3375e78cd65381dbe.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b01751fc2f32ea85a5b52a18bee3a7b.png)
(2)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
您最近一年使用:0次
名校
解题方法
8 . 对于数列
,若满足
(
,p是与n无关的常数),则称数列
是“比等差数列”,常数p称为此数列的“比差”.
(1)已知数列
,
,判断数列
,
是否为“比等差数列”;
(2)证明“比差”为零的“比等差数列”一定是等比数列;
(3)“比差”为正的“比等差数列”是否一定是递增数列?如果是,给出证明;如果不是,请举出反例.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6fa23cea01edcfa9e6a7b1b10ee0f21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(1)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ff1c33b81ac2f065d37faef37504bb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/267c99ff3f6386113dbaa7b1e49612da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)证明“比差”为零的“比等差数列”一定是等比数列;
(3)“比差”为正的“比等差数列”是否一定是递增数列?如果是,给出证明;如果不是,请举出反例.
您最近一年使用:0次
9 . 已知数列
满足
.
(1)证明:数列
是等差数列;
(2)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56f00ce8722a49f404e1dca6d2ed89dc.png)
(1)证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc0b53ddd01ed8617540f85ce89ce82d.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2d6a0d136f2be8c63f966d4da3392ba.png)
您最近一年使用:0次
2022-12-06更新
|
1249次组卷
|
7卷引用:河南省青桐鸣2023届高二上学期11月联考数学试题
河南省青桐鸣2023届高二上学期11月联考数学试题河南省周口市项城市正泰博文学校等3校2022-2023学年高二上学期11月月考数学试题河南省濮阳市2022-2023学年高二上学期期中数学试题安徽省六安第一中学2022-2023学年高二上学期期末数学试题(已下线)拓展三:数列与不等式 -【帮课堂】2022-2023学年高二数学同步精品讲义(人教A版2019选择性必修第二册)(已下线)专题15 数列不等式的证明 微点6 数列不等式的证明综合训练安徽省阜阳市第三中学2023-2024学年高二上学期一调考试(10月月考)数学试题
名校
10 . 已知
和
是各项均为正整数的无穷数列,如果同时满足下面两个条件:
①
和
都是递增数列;
②
中任意两个不同的项的和不是
中的项.
则称
被
屏蔽,记作
.
(1)若
,
.
(i)判断
是否成立,并说明理由;
(ii)判断
是否成立,并说明理由.
(2)设
是首项为正偶数,公差是
的无穷等差数列,判断是否存在数列
,使得
.如果存在,写出一个符合要求的数列
;如果不存在,说明理由;
(3)设
是取值于正整数集的无穷递增数列,且对任意正整数
,存在正整数
,使得
.证明:存在数列
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
则称
![](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/1284d81cf684a54e3070d2c69085c76e.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/677e46ecd051c92489c0d1d458932f37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42fef6975d285cabcf6be67c78f30d30.png)
(i)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1284d81cf684a54e3070d2c69085c76e.png)
(ii)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b5bb0902c0daf52fe26a78a250b96f7.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1284d81cf684a54e3070d2c69085c76e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e33f575cee1cddd9bbc34dcd592a4e2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53442dcf82f93d94f20be6bf2c934cb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1284d81cf684a54e3070d2c69085c76e.png)
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2022-12-05更新
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303次组卷
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2卷引用:北京市海淀区北大附中2023届高三预科部上学期12月阶段练习数学试题