解题方法
1 . 现有甲、乙、丙三个人相互传接球,第一次从甲开始传球,甲随机地把球传给乙、丙中的一人,接球后视为完成第一次传接球;接球者进行第二次传球,随机地传给另外两人中的一人,接球后视为完成第二次传接球;依次类推,假设传接球无失误.
(1)设乙接到球的次数为
,通过三次传球,求
的分布列与期望;
(2)设第
次传球后,甲接到球的概率为
,
(i)试证明数列
为等比数列;
(ii)解释随着传球次数的增多,甲接到球的概率趋近于一个常数.
(1)设乙接到球的次数为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(2)设第
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
(i)试证明数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3d4bdc55e0b2ed6cebd27b8315edadb.png)
(ii)解释随着传球次数的增多,甲接到球的概率趋近于一个常数.
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2 . 若数列
同时满足下列两个条件,则称数列
具有“性质A”.
①
(
);②存在实数
,使得对任意
,有
成立.
(1)设
,试判断
是否具有“性质A”;
(2)设递增的等比数列
的前n项和为
,若
,证明:数列
具有“性质A”,并求出A的取值范围;
(3)设数列
的通项公式
,若数列
具有“性质A”,其满足条件的A的最大值
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e772be971634dc7230df59d91399dc59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/670d5fc71b49fdb5411b046bb9a81bab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/655c46b33730f3a29b9ec3024df71375.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b39b97c1f6007e458646cf2655a0974.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a234044b6e65e53f5f0d979886be4f48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942ed174dfdafaf5f0a68cac579110f8.png)
(2)设递增的等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38ef4c4439b36c2847b0056a116d56d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efe308769d98da3757d3d3c9019ea84e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cce224c28ca451c4f105dc3b077736cb.png)
(3)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d813f3ca8db41a4db6c18eac30fef98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/325bb1f4a88ca4e5bed08b5a4ede97ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d813f3ca8db41a4db6c18eac30fef98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f27f0c9be9710c55ab8e8d2cb4e56df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
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4卷引用:上海市静安区2022届高考二模数学试题
3 . 在现实世界,很多信息的传播演化是相互影响的.选用正实数数列
,
分别表示两组信息的传输链上每个节点处的信息强度,数列模型:
,描述了这两组信息在互相影响之下的传播演化过程.若两组信息的初始信息强度满足
,则在该模型中,关于两组信息,给出如下结论:
①
;
②
;
③
,使得当
时,总有![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2214a97ba3e5cf1fe7b8435fd5241b0.png)
④
,使得当
时,总有
.
其中,所有正确结论的序号是_________
![](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/2ec431d2da21e754e7c528617671ad48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d7c2fa1c39d9ca5aa645e460b770d0b.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/358f2518e1f29a75ff240cd37327b981.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0af55de04e2db91c060b7813fc7f72db.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64a5c77806a86c309544871bf4985872.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52eecd38954cd0ca3fb26328a39bb859.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2214a97ba3e5cf1fe7b8435fd5241b0.png)
④
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64a5c77806a86c309544871bf4985872.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52eecd38954cd0ca3fb26328a39bb859.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6632ab856a0178f547d9aacd9b02a8a3.png)
其中,所有正确结论的序号是
您最近一年使用:0次
2022-05-12更新
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3卷引用:北京市海淀区2022届高三二模数学试题
名校
4 . 下面四个命题中:
(1)若
是等差数列,则
极限不存在;
(2)已知
,当
时,数列
的极限为1或-1;
(3)已知
,则
;
(4)若
,则
,数列
的极限是0;
(5)若
存在,则
的取值范围为
;
(6)若等比数列
的各项和存在,则
.
其中真命题个数为( )
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ec7510dc49bea69b5961573297f1289.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19893fdb67307d35a9115ef4f3f1202a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b103b72c9013ce09ee987c6a7540c0ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9be609302ce8bdfb1b3a91aaadf1c25.png)
(4)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e57d2228252838ca0747e8ef7fa59ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d80e8a674972bdbcc9f85835319b1445.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(5)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb0b45c8ebb4ef6648603c3dad077ebd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14f45aaf3c60f7dc00774e8f9ac3d8b1.png)
(6)若等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c65ab13c42b4866d8e6035a9bc76326.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047056c99b39c70fa40d3c8178e5b631.png)
其中真命题个数为( )
A.1 | B.2 | C.3 | D.4 |
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名校
解题方法
5 . 已知实数列
满足:
,点(
在曲线
上.
(1)当
且
时,求实数列
的通项公式;
(2)在(1)的条件下,若
表示不超过实数t的最大整数,令
,
是数列
的前n项和,求
的值;
(3)当
,
时,若
存在,且
对
恒成立,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e076d53e0fab96afda46ff7ac1689dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdd9a0e365ee8942dda707461757fa2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43982e577f2d2c7e88ad0818d85baa04.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cbeede118c407a800b05757b9a1393e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7aed39f5aca78934fb383402433fe549.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
(2)在(1)的条件下,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77e8eb98a5c6280cf6d56b0ab58b5607.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cd3a1b04fc7cee0560bf0c99579927c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/166a40365a42b31a364defa68c4597b1.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cbeede118c407a800b05757b9a1393e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee138273c0aa7adec692e982085113e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bf08da5d1097370110c2e7c9f52434b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0f3ff0910722efb64fa9a35b14fb9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b7926009be0696df0cf3035661335ca.png)
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名校
解题方法
6 . 若无穷数列{
}满足如下两个条件,则称{
}为无界数列:
①
(n=1,2,3......)
②对任意的正数
,都存在正整数N,使得n>N,都有
.
(1)若
,
(n=1,2,3......),判断数列{
},{
}是否是无界数列;
(2)若
,是否存在正整数k,使得对于一切
,都有
成立?若存在,求出k的范围;若不存在说明理由;
(3)若数列{
}是单调递增的无界数列,求证:存在正整数m,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9645bd4d2002993b90ec6d48f9c04f7.png)
②对任意的正数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/700458c01a7ad031e27d80ed43e9e882.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2a701eb81a1e88c69357f9eae5915ee.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ff1c33b81ac2f065d37faef37504bb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bee912af5e2313d631ff3016ca7cc32d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ff1c33b81ac2f065d37faef37504bb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4fef5f2a4235817fb704d29e08766e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bec3181d9f88a68fb7470d0c9beb183.png)
(3)若数列{
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7407503edea1b02e3084387c8a328d9e.png)
您最近一年使用:0次
2022-03-31更新
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8卷引用:北京市房山区2022届高三一模数学试题
北京市房山区2022届高三一模数学试题(已下线)临考押题卷01-2022年高考数学临考押题卷(北京卷)(已下线)必刷卷03-2022年高考数学考前信息必刷卷(新高考地区专用)(已下线)重难点08 七种数列数学思想方法-2江苏省盐城市2022-2023学年高三上学期期中复习数学试题北京市北师大附属实验中学2021-2022高二下学期数学月考试题北京卷专题18数列(解答题)北京市第五中学2022-2023学年高二下学期期中考试数学试题
7 . 已知数列
的前
项和为
,
,给出以下三个命题:
①
;②
是等差数列;③![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28babe6d68eb0646f9e97bd9e01c3c86.png)
(1)从三个命题中选取两个作为条件,另外一个作为结论,并进行证明;
(2)利用(1)中的条件,证明数列
的前
项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99aed0821e3241d550edfdc7c329cf50.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/b065334d8f60c49f4bd3d9f1373fe4cd.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6e35ba37b0cc1d390e05391554e9660.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99aed0821e3241d550edfdc7c329cf50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28babe6d68eb0646f9e97bd9e01c3c86.png)
(1)从三个命题中选取两个作为条件,另外一个作为结论,并进行证明;
(2)利用(1)中的条件,证明数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fa2c40028103fd23931f8772b0cefa9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c215db1d8f69757118ad405b78035628.png)
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2022-02-22更新
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3卷引用:黑龙江省双鸭山市第一中学2021-2022学年高三上学期期末考试数学(理)试题
解题方法
8 . 已知数列
,
,下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3819d72974cc2440af9a448e5076246a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/794dd235c980222337d01626c96371f9.png)
A.对任意的![]() ![]() ![]() |
B.对任意的![]() ![]() ![]() |
C.对任意的![]() ![]() ![]() |
D.对任意的![]() ![]() ![]() |
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2022-01-03更新
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5卷引用:浙江省绍兴市诸暨市海亮高级中学2022届高三下学期高考前最后一卷数学试题
浙江省绍兴市诸暨市海亮高级中学2022届高三下学期高考前最后一卷数学试题浙江省绍兴市诸暨市海亮高级中学2021-2022学年高三上学期12月选考数学试题(已下线)专题6-1 数列函数性质与不等式放缩(讲+练)-2(已下线)专题9 周期数列 微点2 周期数列的“脸谱”识别(已下线)第4章 数列 章末题型归纳总结(3)
名校
9 . 等差数列
中,公差为
,设
是
的前n项之和,且
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7e0b773ed85bbb06e0d084ca694b7dc.png)
__________ .
![](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/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5892916236834b88bbae412d97eda48a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7e0b773ed85bbb06e0d084ca694b7dc.png)
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2021-10-28更新
|
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3卷引用:课时24 数列的极限与无穷等比数列各项的和-2022年高考数学一轮复习小题多维练(上海专用)
(已下线)课时24 数列的极限与无穷等比数列各项的和-2022年高考数学一轮复习小题多维练(上海专用)上海市上海师范大学附属中学2022届高三下学期3月月考数学试题上海市奉贤区2021届高三上学期一模数学试题
10 . 图形的被覆盖率是指,图形被覆盖部分的面积与图形的原面积之比.通常用字母
表示.如图所示,边长为1的正三角形被
层半径相等的圆覆盖,最下面一层与正三角形底边均相切,每一层相邻两圆外切,层与层相邻的圆相外切,且每一层两侧的圆与正三角形两边相切.记覆盖的等圆层数为
时,等圆的半径为
,
.图中给出
等于1,2,10时的覆盖情形.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/22/8b418677-a88b-4fce-8e18-6293c94b7b24.png?resizew=306)
(Ⅰ)写出
,
的值,并求数列
的通项公式;
(Ⅱ)证明:对任意的层数
,此正三角形的被覆盖率
低于91%.
(参考数据:
,
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2d2adce9c2bb310a074bc39ccc52a3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/22/8b418677-a88b-4fce-8e18-6293c94b7b24.png?resizew=306)
(Ⅰ)写出
![](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/b4be2164a2c67d6163faee87a10942bb.png)
(Ⅱ)证明:对任意的层数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(参考数据:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47d553e4a26eb3012410ef7558a5fd6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/460317e7c26f95b9b29cfe1a89b796d6.png)
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