解题方法
1 . 设数列
是等差数列,且公差为
,若数列
中任意不同的两项之和仍是该数列中的一项,则称该数列是“封闭数列”.
(1)若数列
中,
,
,求证:数列
是“封闭数列”;
(2)若
,试判断数列
是否为“封闭数列”,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(1)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86fc336b4a83bf6d66c4afcc431597f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5c1344592c925b273f2cb9b9e47ebbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf4ae73a6f9c290183bed81c5b622e72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
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2021-09-22更新
|
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|
5卷引用:沪教版(上海) 高二第一学期 新高考辅导与训练 第7章 数列与数学归纳法 7.2(2)等差数列的定义与通项公式的应用
沪教版(上海) 高二第一学期 新高考辅导与训练 第7章 数列与数学归纳法 7.2(2)等差数列的定义与通项公式的应用人教A版(2019) 选修第二册 突围者 第四章 第二节 课时1 等差数列的概念沪教版(2020) 选修第一册 新课改一课一练 第4章 4.1.1 等差数列及其通项公式(已下线)4.1等差数列及其通项公式(第1课时)(作业)(夯实基础+能力提升)-【教材配套课件+作业】2022-2023学年高二数学精品教学课件福建省宁德市2022-2023学年高二上学期居家监测数学试题
2 . 已知数列
,如果数列
满足
,
,则称数列
是数列
的“生成数列”.
(1)若数列
的通项公式为
,写出数列
的生成数列”
的通项公式.
(2)若数列
的通项公式为
(
是常数),则数列
的“生成数列”
是否是等差数列?说明理由.
(3)已知数列
的通项公式为
,求数列
的“生成数列”
的前
项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcd6ea6fbb41eb3d8baab12190778844.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3aaee408bdec05bbdfcd4b841a331e61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bb5c391e084e5c3d3f3fa255d0ec726.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f66502169d7a5f5f36977cc885735d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
(1)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b80c1ed7b10ac7ca1cd81cdd39a8fcc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43b7e7cd571c8cd141cbbfe5d0890bf6.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c88a7ef007c78a93e33bd77c4396626.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efef985a85fb3952fcc8febe3821f540.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c88a7ef007c78a93e33bd77c4396626.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36014b917ba2acefb60dc53fc76ac84a.png)
(3)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11913c4302df9c8bdd8b14a3ac576943.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17577fa9dd28878cecafbabddb0c650d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11913c4302df9c8bdd8b14a3ac576943.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c82537b0d1efd8f5ef044bfdd1d7a224.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
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|
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|
3卷引用:苏教版(2019) 选修第一册 突围者 第4章 易错疑难集训三
名校
3 . 定义数列
如下:
,对任意的正整数
,有
.
(1)写出
,
,
,
的值;
(2)证明:对任意的正整数
,都有
;
(3)是否每一个非负整数都在数列
中出现?证明你的结论.
![](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/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7882fcd2daeb34ad11983155b474cd3c.png)
(1)写出
![](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/f65fc200f10b97588a0c9896277c9c64.png)
(2)证明:对任意的正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/051d406f2e4e9e4232e349d277f58a81.png)
(3)是否每一个非负整数都在数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
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|
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6卷引用:北京市清华大学附属中学2020-2021学年高二下学期期中数学试题
北京市清华大学附属中学2020-2021学年高二下学期期中数学试题(已下线)第4章 数列 单元综合检测(重点)(单元培优)-2021-2022学年高二数学课后培优练(苏教版2019选择性必修第一册)(已下线)4.4 数学归纳法(课堂培优)-2021-2022学年高二数学课后培优练(苏教版2019选择性必修第一册)(已下线)4.4 数学归纳法-2022-2023学年高二数学《基础·重点·难点 》全面题型高分突破(苏教版2019选择性必修第一册)(已下线)2020年高考北京数学高考真题变式题16-21题北京市十一学校2022届高三4月月考数学试题
4 . 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)
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名校
5 . 若数列
满足:对于任意的
,总存在
且
,使
成立,则称数
列为“Z数列”.
(1)若
,判断数列
是否为“Z数列”,说明理由;
(2)证明等差数列
为“Z数列”的充要条件是“
的公差d等于首项
”;
(3)是否存在既是等比数列又是“Z数列”的数列
?若存在,求出所有可能的公比的值,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/0d60e9f844afc9e372d0112bb2279214.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/738d701bf35c715a18b1e917d188a115.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/814698e07a0447e71fc9cb91504fcc33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3c5b67b7e8d7d71d6ca7875d4c2de6c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/677e46ecd051c92489c0d1d458932f37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)证明等差数列
![](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/e72adb45c60c2f63b46e65ff787302bf.png)
(3)是否存在既是等比数列又是“Z数列”的数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
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5卷引用:沪教版(2020) 选修第一册 同步跟踪练习 第4章 测试卷
沪教版(2020) 选修第一册 同步跟踪练习 第4章 测试卷上海市格致中学2021届高三三模数学试题(已下线)课时22 数列、等差数列、等比数列-2022年高考数学一轮复习小题多维练(上海专用)(已下线)2021年新高考北京数学高考真题变式题16-21题北京市第五十五中学2023届高三上学期10月月考数学试题
名校
6 . 对于数列
,定义
为数列
的差分数列,其中
.如果对任意的
,都有
,则称数列
为差分增数列.
(1)已知数列
为差分增数列,求实数
的取值范围;
(2)已知数列
为差分增数列,且
,
.若
,求非零自然数k的最大值;
(3)已知项数为2k的数列
(
)是差分增数列,且所有项的和等于k,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5a12a1692b36e4bf3a867220d099e31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a87ee5c855af8543cd8b87bb009a869b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/deda945164283569437cda6976fe35ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b05563cb4df28aa2083ec58142e3f4af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
(1)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72ac34845fee87c6db7afdf743346503.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(2)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/736dc8ce0e8bf2f0cc7cc8b42d6b623b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42bfe87a8f66a72d24ffd73e36f2e430.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1998ec6504bc2f5d43c1e7f0c8f69284.png)
(3)已知项数为2k的数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0208908fa748f1c3acd7ea969646392c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eaa5533405e21f9550df95b8f50cb1ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b424900b9460b02d3559bfc8df1abc44.png)
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|
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6卷引用:专题10 数列(难点)-2020-2021学年高二数学下学期期末专项复习(北师大版2019选择性必修第一册、第二册)
(已下线)专题10 数列(难点)-2020-2021学年高二数学下学期期末专项复习(北师大版2019选择性必修第一册、第二册)(已下线)第4章 数列 单元综合检测(难点)(单元培优)-2021-2022学年高二数学课后培优练(苏教版2019选择性必修第一册)上海市松江区第四中学2022-2023学年高二上学期期中数学试题上海市崇明区2021届高三二模数学试题(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)上海市洋泾中学2023届高三上学期10月月考数学试题
7 . 已知数列
的前
项和为
,把满足条件
的所有数列
构成的集合记为
.
(1)若数列
的通项为
,则
是否属于
?
(2)若数列
是等差数列,且
,求
的取值范围;
(3)若数列
的各项均为正数,且
,数列
中是否存在无穷多项依次成等差数列,若存在,给出一个数列{an}的通项:若不存在,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/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c9600039e8f4d6f73e37c53817209d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(1)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a78a26e3eeac053424c52ab90f6a3490.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eab502f4746976616a116def2ad9f860.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
(3)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52a64e8b2d22c198754636228638bbd3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c331999c6070ea778ce3a0b6e967b16f.png)
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8 . 已知
是无穷数列.给出两个性质:①对于
中任意两项
,在
中都存在一项
,使得
;②对于
中任意项
,在
中都存在两项
,使得
.
(1)若
,判断数列
是否满足性质①,说明理由;
(2)若
,判断数列
是否同时满足性质①和性质②,说明理由;
(3)若
是递增数列,
,且同时满足性质①和性质②,证明:
为等差数列.
.
![](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/dd47818a20119bd6fb1a708d7225cb86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/681ae1522a36768618f7ddaf74abbb7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f2f3c848b79128af6478031cd7ee97d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf16339dca6781c6a4ad485c4b5a04e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb42075543388438384084900b95df48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4350546edf19a072f4e4dd197a740b8.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d148a17ee4d3a7e7027bf1384d093eec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97163015df118267daa64c7a00180ebe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1bae03ee4ac75dacfb026290e4207dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
.
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2021-04-10更新
|
665次组卷
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8卷引用:北京市东城区2020-2021学年高二上学期期末考试数学试题
北京市东城区2020-2021学年高二上学期期末考试数学试题北京市第八十中学2020-2021学年高二下学期期中考试数学试题(已下线)专题04 《数列》中的解答题压轴题(1)-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)上海市南洋模范中学2021-2022学年高二上学期期末数学试题北京市第一七一中学2023-2024学年高二上学期12月月考数学试题(已下线)第四章 数列(单元重点综合测试)-2023-2024学年高二数学单元速记·巧练(苏教版2019选择性必修第一册)北京市顺义区2021届高三上学期期末考试数学试题北京市顺义区第一中学2024届高三上学期12月月考数学试题
2021高三·江苏·专题练习
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9 . 若对于数列{an}中的任意两项ai,aj(i>j),在{an}中都存在一项am,使得am=
,则称数列{an}为“X数列”,若对于数列{an}中的任意一项an(n≥3),在{an}中都存在两项ak,al(k>l),使得an=
,则称数列{an}为“Y数列”.
(1)若数列{an}为首项为1公差也为1的等差数列,判断数列{an}是否为“X数列”,并说明理由;
(2)若数列{an}的前n项和Sn=2n﹣1(n∈N*),求证:数列{an}为“Y数列”;
(3)若数列{an}为各项均为正数的递增数列,且既为“X数列”,又为“Y数列”,求证:a1,a2,a3,a4成等比数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9357edd9952c944763ff3505881d59e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86e284fbea9ea834ede9a9d1b933c1b2.png)
(1)若数列{an}为首项为1公差也为1的等差数列,判断数列{an}是否为“X数列”,并说明理由;
(2)若数列{an}的前n项和Sn=2n﹣1(n∈N*),求证:数列{an}为“Y数列”;
(3)若数列{an}为各项均为正数的递增数列,且既为“X数列”,又为“Y数列”,求证:a1,a2,a3,a4成等比数列.
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10 . 任取一个正整数,若是奇数,就将该数乘3再加上1;若是偶数,就将该数除以2.反复进行上述两种运算,经过有限次步骤后,必进入循环圈1→4→2→1.这就是数学史上著名的“冰雹猜想”(又称“角谷猜想”等).如取正整数
,根据上述运算法则得出6→3→10→5→16→8→4→2→1,共需经过8个步骤变成1(简称为8步“雹程”).
现给出冰雹猜想的递推关系如下:已知数列
满足:
(m为正整数),
.
(1)当
时,试确定使得
需要多少步雹程;
(2)若
,求m所有可能的取值集合M.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3967d620e2fef3ecc724c66e29f68a8.png)
现给出冰雹猜想的递推关系如下:已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/999ac8c1ef39251e07a7fc54cbf7e26e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c34aebde5a43d798d462d7da3073d1f.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d22d1aef8a6e91ddd52cbf1f5b3f32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e15ffa7fecea3704dc892ea8cd513c59.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a15c1d9819e7beecc90744323b0063e.png)
您最近一年使用:0次
2021-02-07更新
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1910次组卷
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4卷引用:人教A版(2019) 选择性必修第二册 新高考名师导学 第四章 复习参考题4