1 . 数列
:
,
,
,…,
,…,对于给定的
(
,
),记满足不等式:
(
,
)的
构成的集合为
.
(Ⅰ)若数列
,写出集合
;
(Ⅱ)如果
(
,
)均为相同的单元素集合,求证:数列
,
,…,
,…为等差数列;
(Ⅲ)如果
(
,
)为单元素集合,那么数列
,
,…,
,…还是等差数列吗?如果是等差数列,请给出证明;如果不是等差数列,请给出反例.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.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/291c25fc6a69d6d0ccfb8d839b9b4462.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/544530e1133b2924ccfbe691141a5641.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60939f5f5cd85a28dcb63d2f78d26b60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2eb18547717a019d4b546b8dd0b0365c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/030137376417efb2ac10443ff54fbfb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13c0b3d5b60308da39aaf5493d58f444.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe779a0f086e93f260a1b0c9be9cc415.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e2ea3d72f46edfb7216c7bc9ab9cf9a.png)
(Ⅰ)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f41c721ebc7a5f8346da3c44af85a047.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7e612765b49f8cdda75bdaaf4f86edd.png)
(Ⅱ)如果
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e2ea3d72f46edfb7216c7bc9ab9cf9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60939f5f5cd85a28dcb63d2f78d26b60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/544530e1133b2924ccfbe691141a5641.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/3282e5fde4ae53fcb1bb072a685304c9.png)
(Ⅲ)如果
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e2ea3d72f46edfb7216c7bc9ab9cf9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60939f5f5cd85a28dcb63d2f78d26b60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/544530e1133b2924ccfbe691141a5641.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/3282e5fde4ae53fcb1bb072a685304c9.png)
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解题方法
2 . 若存在常数
,使对任意的
,都有
,则称数列
为
数列.
(1)已知
是公差为2的等差数列,其前n项和为
.若
是
数列,求
的取值范围;
(2)已知数列
的各项均为正数,记数列
的前n项和为
,数列
的前n项和为
,且
.
①求证:数列
是等比数列;
②设
,试证明:存在常数
,对于任意的
,数列
都是
数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcd9218a657b17654c5d757a6f7dee9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba5aa7d54a7d5f2d8fcad75ea832c57c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63455d28754568623491a21003ef0c42.png)
(1)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0519af3f19dde038fab2e68b5e2a5387.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/490c60c8a45e851a00b3d1fd03a96500.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
(2)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43b7e7cd571c8cd141cbbfe5d0890bf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43b7e7cd571c8cd141cbbfe5d0890bf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e15526f7c892333030073b85fc3baee6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc0b178ad283aae528156da949fafd68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a93c798bb1e9f80e452af86b3d231131.png)
①求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43b7e7cd571c8cd141cbbfe5d0890bf6.png)
②设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7a7ca771cef2edbbc26377d3719d266.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcd9218a657b17654c5d757a6f7dee9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd4a2ae3acc076ef0a942c37edb8ce07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c88a7ef007c78a93e33bd77c4396626.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63455d28754568623491a21003ef0c42.png)
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3 . 已知数列
满足
,
.记
,其中
表示不超过m的最大整数,求
的值为____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ea8d0e50065114b05ef2dc1ea1129cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9622d436ef65ed61cc15e822459f77c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25814fb33f1aa4432ee6c820cd45b84c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23145dde872b37ef225d66504b43e21b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/453e398c6f4b0f5247906161e084ed2d.png)
您最近一年使用:0次
4 . 设数列
中,若
,则称数列
为“凸数列”.
(1)设数列
为“凸数列”,若
,
,试写出该数列的前6项,并求出该6项之和;
(2)在“凸数列”中,求证
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45c06715eb0e30cb356c02c9559bca8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(1)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3299d1d394efc1381671b1632e6e87e1.png)
(2)在“凸数列”中,求证
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebd5d6d3e493f202ce9f9d05a02f8eb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
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名校
5 . 设正整数数列
满足
.
(1)若
,请写出所有可能的
的取值;
(2)求证:
中一定有一项的值为1或3;
(3)若正整数m满足当
时,
中存在一项值为1,则称m为“归一数”,是否存在正整数m,使得m与
都不是“归一数”?若存在,请求出m的最小值;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/d7138cc707c7bec7c0c5a60ffd25c1c2.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/259b2e755105c0ee479eabf7265a76a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)若正整数m满足当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/999ac8c1ef39251e07a7fc54cbf7e26e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0623207595425920f16e76a7f8f268b6.png)
您最近一年使用:0次
2020-02-23更新
|
669次组卷
|
3卷引用:2020届北京市十一学校高三(12月)月考数学试题
解题方法
6 . 用
表示一个小于或等于
的最大整数.如:
,
,
. 已知实数列
、
、
对于所有非负整数
满足
,其中
是任意一个非零实数.
(Ⅰ)若
,写出
、
、
;
(Ⅱ)若
,求数列
的最小值;
(Ⅲ)证明:存在非负整数
,使得当
时,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3420606c96b68fb884c839923fd20a25.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53e2f5e82abde22ee1dae0c6d73e32d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04d0024c22d248668a379d8dd1b84cad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f35f7dcce39f3d4dc6b7faf84dc1d0a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96377872af787cbf0d0d24d1db455f75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f35f7dcce39f3d4dc6b7faf84dc1d0a1.png)
(Ⅰ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47767a5d0f95d7d936aa664e71b52baa.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)
(Ⅱ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd65631a58898fc1919ce8b4892835b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8aaf6e3d7a39c0583e58599c05d86f7d.png)
(Ⅲ)证明:存在非负整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52608aa7220d68349e5bc5659072693a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d29f1962d214d7510b868a3a5055e41.png)
您最近一年使用:0次
名校
7 . 设数列
的前
项和是
,令
,称
为数列
,
,…,
的“理想数”,已知数列
,
,…,
的“理想数”为2012,则数列6,
,
,…,
的理想数为( )
![](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/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/945d3a467b1c212578437f50146f56fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.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/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/202900336e3bcd823953fbccf02daaef.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/202900336e3bcd823953fbccf02daaef.png)
A.2014 | B.2015 | C.2016 | D.2017 |
您最近一年使用:0次
2021-03-22更新
|
470次组卷
|
4卷引用:上海市黄浦区格致中学2015-2016学年高二上学期第二次测验数学试题
上海市黄浦区格致中学2015-2016学年高二上学期第二次测验数学试题(已下线)专题17 数学中的新定义问题-2021年高考冲刺之二轮专题精讲精析江西南昌青山湖区南昌三中雷式学校2020-2021学年高一下学期期中数学试题河北省深州市长江中学2021-2022学年高二上学期期末数学试题
名校
8 . 定义“等积数列”:如果一个数列从第2项起,每一项与它的前一项的乘积都等于同一个不为零的常数,那么这个数列叫做等积数列,这个常数叫做等积数列的公积.已知数列
是
,公积为
的等积数列,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbfc875ca919921e8f63a6fca648561b.png)
______ ;数列
的前
项和![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04c2864e2ec3416cc4c081ac1f71a0af.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039e4fe671d61e59b96ee525c9df43e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01317332a203c898536b1d0459f51d23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbfc875ca919921e8f63a6fca648561b.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/04c2864e2ec3416cc4c081ac1f71a0af.png)
您最近一年使用:0次
2020-03-02更新
|
585次组卷
|
4卷引用:专题02 过“三关”破解数列新情境问题 (第三篇)-2020高考数学压轴题命题区间探究与突破
(已下线)专题02 过“三关”破解数列新情境问题 (第三篇)-2020高考数学压轴题命题区间探究与突破北京市大兴区2019-2020学年高二第一学期期中考试数学试题甘肃省会宁县第一中学2020-2021学年高二上学期期中考试数学(文)试题(已下线)专题13 等积数列 微点1 等积数列常见问题
9 . 若有穷数列
满足
且对任意的
,
至少有一个是数列
中的项,则称数列
具有性质![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(1)判断数列1,2,4,8是否具有性质P,并说明理由;
(2)设项数为
的数列
具有性质
,求证:
;
(3)若项数为
的数列
具有性质
,写出一个当
时,
不是等差数列的例子,并证明当
时,数列
是等差数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5b96f565b4ca625ab41a782e3dfd0f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0492686dc1959ba361d9b2832491620.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e72ad2e72453867d089770c3f4c63da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(1)判断数列1,2,4,8是否具有性质P,并说明理由;
(2)设项数为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3411ddca520e2bcb516d0c5c0832aeb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee349b3f104aa5a5e03830a205570f3.png)
(3)若项数为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3411ddca520e2bcb516d0c5c0832aeb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcbd5bb726a08c308b48373afebbb768.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0266e0e890fb1b84be352fdc65bb298.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
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2020-12-25更新
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587次组卷
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6卷引用:重难点01 数列(基本通项求法)-2021年高考数学【热点·重点·难点】专练(上海专用)
(已下线)重难点01 数列(基本通项求法)-2021年高考数学【热点·重点·难点】专练(上海专用)(已下线)考向17 数列新定义-备战2022年高考数学一轮复习考点微专题(上海专用)(已下线)专题05 《数列》中的解答题压轴题(2)-2021-2022学年高二数学同步培优训练系列(苏教版2019选择性必修第一册)上海市嘉定区2021届高三上学期一模数学试题北京市第五十五中学2022-2023年高二下学期3月调研数学试题(已下线)上海高二下学期期末真题精选(压轴60题35个考点专练)-【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)
名校
解题方法
10 . 如果一个数列
满足
(H为常数,
),则称数列
为等和数列,H为公和,
是其前n项的和,已知等和数列
中,
,
,则
等于( )
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
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