1 . 已知
是无穷数列,对于k,
,给出三个性质:
①
(
);
②
(
);
③
(
)
(1)当
时,若
(
),直接写出m的一个值,使数列
满足性质②,若满足求出
的值;
(2)若
和
时,数列
同时满足条件②③,证明:
是等差数列;
(3)当
,
时,数列
同时满足条件①③,求证:数列
为常数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bd2c3166d0bfd9e64bdc85081445e95.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d908582b5cb7fe6ac42e30b01fcc0d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7247567230a3bebb8fa497c2b22bb02.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/422d77af5246812f6e8a67374c8a1b5e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7247567230a3bebb8fa497c2b22bb02.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1154b73b3eeaf33da8dfe0cf88e2ec64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7247567230a3bebb8fa497c2b22bb02.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8e69866076dcff686a05e9e91e61e68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72fdfc045c642fc60935c663da11cc5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7247567230a3bebb8fa497c2b22bb02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/714debb9497560bb3f3eb6e21e75995d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e934a957d0b038f79a2f47415edba01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8e69866076dcff686a05e9e91e61e68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/060e7930731eddbcfac592b808e9b698.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
您最近一年使用:0次
2024-03-12更新
|
460次组卷
|
2卷引用:北京市平谷区2023-2024学年高三下学期质量监控(零模)数学试卷
2023高三·全国·专题练习
解题方法
2 . 在数列
中,已知
,且
.
(1)用数学归纳法证明:
;
(2)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1adb803e408070ec2d1e1063d619aaf9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58caa28b5eb295f163c195ae6d28f46c.png)
(1)用数学归纳法证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f04f01a161c7216cf5b664f5963b0a27.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f8a1328993ec73318e5fc575405a7af.png)
您最近一年使用:0次
名校
3 . 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: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
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)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)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)(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
20-21高二上·全国·单元测试
解题方法
4 . 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
解题方法
5 . 已知数列
中,
.
(Ⅰ)证明:
;
(Ⅱ)证明:
;
(Ⅲ)设数列
的前
项和为
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c99e4abbb741fe508d26891b0bd5193.png)
(Ⅰ)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2798e1dcab1f7f0fe3b8a94b3cd6a88.png)
(Ⅱ)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69d9d01228619eed5d540c35493137fc.png)
(Ⅲ)设数列
![](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/686184c721e3e47a08fad5fe5a60f9b5.png)
您最近一年使用:0次
6 . 设集合
由满足下列两个条件的数列
构成:①
②存在实数
使
对任意正整数
都成立.
(1)现在给出只有5项的有限数列
其中
;
试判断数列
是否为集合
的元素;
(2)数列
的前
项和为
且对任意正整数
点
在直线
上,证明:数列
并写出实数
的取值范围;
(3)设数列
且对满足条件②中的实数
的最小值
都有
求证:数列
一定是单调递增数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91edc7e2d4811f5ea6c01284cf00393a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c9bcdee28896b41b8d8b5460e3317f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f96ecb5f9618e6488e0459c8efcd00c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(1)现在给出只有5项的有限数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b86716bf332a140dc65315f0f7974d9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3850dacad99ef11d24c0010988897e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7ba47afbed1f1c5813ed960b4c83dea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6dcba920904a03e3f950e962cc8c7ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91edc7e2d4811f5ea6c01284cf00393a.png)
(2)数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c88a7ef007c78a93e33bd77c4396626.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb9c41a7e8200dbea0e52aa1c9dbd022.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51e2283bd51dff58ab4ee5b5fd01f451.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c935db1c9f12fc8fc240ed1c8540736b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9450684ca4c5f6ee21f226f5f40abe2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fa268fa52d6e44fe9c4bd3c2580f1b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
(3)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2a8451d52f8aee6156952a8c2872bec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7733868f355470241f885ce7a25d2012.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a4efa719ef47c04135d959c9024fe47.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11913c4302df9c8bdd8b14a3ac576943.png)
您最近一年使用:0次
7 . 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
8 . 如果数列
对任意的
满足:
,则称数列
为“
数列”.
(1)已知数列
是“
数列”,设
,求证:数列
是递增数列,并指出
与
的大小关系(不需要证明);
(2)已知数列
是首项为
,公差为
的等差数列,
是其前
项的和,若数列
是“
数列”,求
的取值范围;
(3)已知数列
是各项均为正数的“
数列”,对于
取相同的正整数时,比较
和
的大小,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff42d971347ba9bdfb7b88edaab14861.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(1)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22baa00f7f114d977b3db7f6e6b4a5be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aeeecf37e31620750230959b9694bdf7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5473743f8da0d22004b2693e75a28198.png)
(2)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57124599d0a72bf788f08ce442a85600.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13aababf2d1ee2f7c5336857f0fcb7a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
(3)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbbd81dde229100be2244958fe333952.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942d129c0f29f54819024c5b774f6525.png)
您最近一年使用:0次
9 . 一个大于1的自然数,除了1和它本身外,不能被其他自然数整除,则称这个数为质数.质数的个数是无穷的.设由所有质数组成的无穷递增数列
的前
项和为
,等差数列1,3,5,7,…中所有不大于
的项的和为
.
(Ⅰ)求
和
;
(Ⅱ)判断
和
的大小,不用证明;
(Ⅲ)设
,求证:
,
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/960b682f983b053dc9064cf29c97e250.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/3ffb021aa7d5a5c2f0691e337caad624.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38fcec7af3520884b173b29bda6c657a.png)
(Ⅰ)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e727bd61ff43fe654f6532abe939b85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3627e4ccde7d69c49034a4a2d10bee5.png)
(Ⅱ)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38fcec7af3520884b173b29bda6c657a.png)
(Ⅲ)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05becbd3c66ebe95299916c1c2279392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5141a5b907f5ff11bbd7cacbd7b5db3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31c8a8e9b48cd1a519793b3fd840a522.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d7024231adcecb545198697a2037efe.png)
您最近一年使用:0次
10 . 设数列
的各项均为不等的正整数,其前
项和为
,我们称满足条件“对任意的
,均有
”的数列
为“好”数列.
(1)试分别判断数列
,
是否为“好”数列,其中
,
,
,并给出证明;
(2)已知数列
为“好”数列.
① 若
,求数列
的通项公式;
② 若
,且对任意给定正整数
(
),有
成等比数列,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.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/7a1b5d155760aea9f29fe3a3a9034bc5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2304c0072be971b5b8933a680d6a70af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
(1)试分别判断数列
![](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/677e46ecd051c92489c0d1d458932f37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7bb72b3ebbca741b3eda49cd617c058.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
(2)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38ef4c4439b36c2847b0056a116d56d4.png)
① 若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7eedd069693fa76f371c8205d026c957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38ef4c4439b36c2847b0056a116d56d4.png)
② 若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ff2eb0fb845a82db057a3bbaf314c5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4fce4202f83d6ccf98640d31a734a18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f6119b6def1776f745aa5d7b9c00701.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2490e20d553e4b6e85621ca905fb3a1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cbdd68463267382ef3d410eb5417a23.png)
您最近一年使用:0次
2018-10-23更新
|
693次组卷
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4卷引用:江苏省徐州市2019届高三第一学期期中模拟试卷数学
江苏省徐州市2019届高三第一学期期中模拟试卷数学江苏省南通市2020届高三下学期6月模拟考试数学试题(已下线)专题6.1 数列的概念与简单表示法(练)-江苏版《2020年高考一轮复习讲练测》(已下线)专题6.3 等比数列及其前n项和(练)-江苏版《2020年高考一轮复习讲练测》