2024高三·全国·专题练习
1 . 在数列
中,
,
.求证:
为等差数列;
![](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/c9dbac8ba528f6cc358553bf0d757204.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1fda4ad3d9f8b74cca6a95fa5a84f19.png)
您最近一年使用:0次
2 . 第19届亚运会于2023年9月23日至10月8日在杭州举行,为弘扬奥林匹克和亚运精神,增强锻炼身体意识,某学校举办一场羽毛球比赛.已知羽毛球比赛的单打规则是:若发球方胜,则发球方得1分,且继续在下一回合发球;若接球方胜,则接球方得1分,且成为下一回合发球方.现甲、乙二人进行羽毛球单打比赛,根据以往甲、乙两名运动员对阵的比赛数据可知,若甲发球,甲得分的概率为
,乙得分的概率为
;若乙发球,乙得分的概率为
,甲得分的概率为
.规定第1回合是甲先发球.
(1)求第3回合由甲发球的概率;
(2)①设第i回合是甲发球的概率为
,证明:
是等比数列;
②已知:若随机变量
服从两点分布,且
,
,2,…,n,则
.若第1回合是甲先发球,求甲、乙连续进行n个回合比赛后,甲的总得分的期望.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eac97e6740365c85ad857aff85cefbe5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d33adb74906403b0b00fcbd9fa691d8b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7294f5ae2a24ff42e84cd9773b2a7287.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3ffd5c35bba71ea54c28622b6cf505d.png)
(1)求第3回合由甲发球的概率;
(2)①设第i回合是甲发球的概率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb5c607987b73502db63f77c9799f4bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74bfab5f9cb89603b6313c971285ff3b.png)
②已知:若随机变量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f95e54a9b7c66c97dc6ee6161a25c0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86ef015dfcb4e200426d5f54ba6deec4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c45176df950dfe48b8ca7eac08ee349.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ece16154c3be9e43a5dd37a91d7d8c3b.png)
您最近一年使用:0次
名校
解题方法
3 . 为了避免就餐聚集和减少排队时间,某校食堂从开学第1天起,每餐只推出即点即取的米饭套餐和面食套餐.某同学每天中午都会在食堂提供的两种套餐中选择一种套餐,如果他第1天选择了米饭套餐,那么第2天选择米饭套餐的概率为
;如果他第1天选择了面食套餐,那么第2天选择米饭套餐的概率为
.已知他开学第1天中午选择米饭套餐的概率为
.
(1)求该同学开学第2天中午选择米饭套餐的概率;
(2)记该同学第
天选择米饭套餐的概率为
,
(i)证明:
为等比数列;
(ii)证明:当
时,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf31876698721a199c7c53c6b320aa86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf31876698721a199c7c53c6b320aa86.png)
(1)求该同学开学第2天中午选择米饭套餐的概率;
(2)记该同学第
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfc321599521a98661ed719cc82ca87c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf83e20035c3afd6d26ebfd53d768a70.png)
(i)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7df0c50dc04f2e056ccf81192a00de24.png)
(ii)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d42c5e562fa5fa8cf3759b52cd0c139.png)
您最近一年使用:0次
2024-01-26更新
|
1715次组卷
|
6卷引用:第4讲:概率与数列的结合问题【讲】
(已下线)第4讲:概率与数列的结合问题【讲】(已下线)题型27 5类概率统计大题综合解题技巧(已下线)专题3.5马尔科夫链模型(强化训练)-2023-2024学年高二数学下学期重难点突破及混淆易错规避(人教A版2019)山西省太原市2024届高三上学期期末学业诊断数学试题浙江省嘉兴市第一中学2024届高三第一次模拟测试数学试题福建省莆田第四中学2023-2024学年高二下学期期中考试数学试卷
4 . 已知数列
满足
,
,
,求
的通项公式.
![](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/ca4123422b5a6621da6a3214aa8c3e2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a1bff6028f612801eab56c6c73be073.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
您最近一年使用:0次
5 . 对于给定的数列
,如果存在实常数
,使得
对于任意
都成立,我们称数列
是“优美数列”.
(1)若
,数列
是否为“优美数列”?若是,指出它对应的实常数
,若不是,请说明理由;
(2)已知数列
满足
.若数列
是“优美数列”,求数列
的通项公式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a550c916c64f621010e604a30ef67566.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5306f3f7463bfbe4fd492cabd187dee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/930bc56406e69b785b37a83d48e36724.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32b1708393d9072a03549fb05f2f7d4b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0914c295f572c98dd043d4f84268934.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a550c916c64f621010e604a30ef67566.png)
(2)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/712b15640ea3f3f2170203ba08b50261.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
名校
解题方法
6 . 已知数列
,若
为等比数列,则称
具有性质P.
(1)若数列
具有性质P,且
,
,求
的值;
(2)若
,求证:数列
具有性质P;
(3)设
,数列
具有性质P,其中
,
,
,若
,求正整数m的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f03b007be99a17613246b5ea1ff86d75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa0dc13236eaa2bd0cdc0f24beea11fe.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/8323901a49cac29afd7d62864f088077.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ced4e381e8c3336848b8c436dbc584f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daf464629fa321a6ff7401ab79f07083.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7148956a39b0ef8d2cff51ea3e71d06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/864fb22e698e7595dc8c8aaa7cd1d83b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dea6578afabc23f5d7041b88c3790dd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4afcfe474c77ea823488bee2c0a3bf0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efd92c8f97571daf32d174e58cb14926.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b363aef37c2a1823ee68a9046b1dec3f.png)
您最近一年使用:0次
2024-01-15更新
|
451次组卷
|
6卷引用:第4章 数列(压轴题专练)-2023-2024学年高二数学单元速记·巧练(沪教版2020选择性必修第一册)
(已下线)第4章 数列(压轴题专练)-2023-2024学年高二数学单元速记·巧练(沪教版2020选择性必修第一册)上海市北虹高级中学2023-2024学年高二上学期期末数学试题福建省莆田市第二十五中学2023-2024学年高二上学期期末数学试题(已下线)专题01 数列(九大题型+优选提升题)-【好题汇编】备战2023-2024学年高二数学下学期期末真题分类汇编(沪教版2020选择性必修,上海专用)辽宁省沈阳市东北育才学校2023-2024学年高二实验部下学期阶段检测二(6月)数学试题上海市闵行区六校期末联考2023-2024学年高一下学期6月期末考试数学试题
名校
解题方法
7 . 约数,又称因数.它的定义如下:若整数
除以整数
得到的商正好是整数而没有余数,我们就称
为
的倍数,称
为
的约数.设正整数
共有
个正约数,即为
,
,
,
,
.
(1)当
时,若正整数
的
个正约数构成等比数列,请写出一个
的值;
(2)当
时,若
,
,
,
构成等比数列,求正整数
的所有可能值;
(3)记
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87484a879f450ab097f720fb2a0f4a2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.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/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68c0cd13ec90e5697013e59d73d3e82c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afeed05dbd9752dd537a06bbcbc867cf.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcbd5bb726a08c308b48373afebbb768.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbeaed9ec21e090defafcfeefe0059c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe164d8a8a4049e01565b576007651de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01416ee1d48b17f889e444b7eda99740.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b95a49832d7c33597639bea9eace7989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a57e391b1d575796894fea80cce6329b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04bc6dcaef3c78886e21f1c41e7f2cd6.png)
您最近一年使用:0次
2024-05-04更新
|
167次组卷
|
12卷引用:高考数学冲刺押题卷02(2024新题型)
(已下线)高考数学冲刺押题卷02(2024新题型)(已下线)微考点4-1 新高考新试卷结构压轴题新定义数列试题分类汇编(已下线)专题06 数列湖南省长沙市雅礼中学2024届高三一模数学试卷(已下线)第四套 艺体生新高考全真模拟 (一模重组卷)湖南省常德市第一中学2023-2024学年高二下学期第一次月考数学试题北京市西城区北京师范大学第二附属中学2023-2024学年高二下学期期中考试数学试题(已下线)高二下学期第三次月考模拟卷(新题型)(范围:导数+选择性必修第三册)-2023-2024学年高二数学题型分类归纳讲与练(人教A版2019选择性必修第三册)北京市通州区2023届高三上学期期末数学试题北京市第五十五中学2024届高三上学期10月月考数学试题北京市东城区第六十五中学2024届高三上学期12月月考数学试题广东省广州市广东实验中学2023-2024学年高三下学期教学情况测试(二)数学试卷A
8 . 在数列
中,
,
,且数列
是等比数列.
(1)求
的通项公式;
(2)设
,数列
的前n项和为
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14835bf3f00139ccec0694d0924db795.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76d4a32c09309e0fc8eb7576e0742bec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bf3da897eb73b729f66bb0d700775c5.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2366d8d61a81a296a898fc50d8db6d75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e9135a6c67cf88b814b5781276f8809.png)
您最近一年使用:0次
9 . 在数列
中,已知
.
(1)求
的通项公式;
(2)求数列
的前
项和.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5156656bf7d5d7da58f2e8e220bec754.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce07afa90e76a49bae2a0e1c0f58414b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
您最近一年使用:0次
2024-01-10更新
|
1316次组卷
|
5卷引用:模块六 大招1 一阶线性递推
(已下线)模块六 大招1 一阶线性递推(已下线)第2讲:复杂数列通项和求和【练】河南省周口市项城市2024届高三上学期1月阶段测试数学试题陕西省安康市高新中学、安康中学高新分校2024届高三上学期第二次“尖子生计划”考试文科数学试题陕西省安康市高新中学、安康中学高新分校2024届高三上学期第二次“尖子生计划”考试理科数学试题
10 . 已知数列
满足
,
,且
.
(1)求证:数列
为等比数列;
(2)若
,求数列
的前n项的和
.
![](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/2693734765399876e9e93cdb110231c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d292da83f1502449e6118c83e4a94d5f.png)
(1)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d82c65a855b1eed9c43e6829f6c3bffb.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e34f2503b45b54100a1c9f9b000860c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
您最近一年使用:0次