名校
解题方法
1 . 根据统计数据,某种植物感染病毒之后,其存活日数X满足:对于任意的
,
的样本在
的样本里的数量占比与
的样本在全体样本中的数量占比相同,均等于
,即
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4111fbeeb20975034f051295bc7bb1f2.png)
__________ ,设
,
的前n项和为
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04c2864e2ec3416cc4c081ac1f71a0af.png)
___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a05d83daef7bb72f157ed504d3a1708c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7bbadf6bea687b3de5dd1afe160a020.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71ce9db5574a2df6184bdc7cd13b208a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3ffd5c35bba71ea54c28622b6cf505d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fc7e207019873e3b048a13d4876c952.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4111fbeeb20975034f051295bc7bb1f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ea3df8f05aab080084fe846c692c321.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04c2864e2ec3416cc4c081ac1f71a0af.png)
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3卷引用:辽宁省大连市第二十三中学2024届高三下学期校模拟考试数学试题
2 . 随着科技的发展,越来越多的智能产品深入人们的生活.为了测试某品牌扫地机器人的性能,开发人员设计如下实验:如图,在
表示的区域上,扫地机器人沿着三角形的边,从三角形的一个顶点等可能的移动到另外两个顶点之一,记机器人从一个顶点移动到下一个顶点称执行一次程序.若开始时,机器人从
点出发,记机器人执行
次程序后,仍回到
点的概率为
,则下列结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/019e032a954865d46eb35956a8fc2fe3.png)
A.![]() | B.![]() ![]() |
C.![]() | D.![]() |
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2卷引用:辽宁省2023-2024学年高二下学期期初教学质量检测数学试题
解题方法
3 . 踢毽子在我国流传很广,有着悠久的历史,是一项传统民间体育活动.某次体育课上,甲、乙、1丙、丁四人一起踢毽子.毽子在四人中传递,先从甲开始,甲传给乙、丙、丁的概率均为
;当乙接到毽子时,乙传给甲、丙、丁的概率分别为
,
,
;当丙接到毽子时,丙传给甲、乙、丁的概率分别为
,
,
;当丁接到毽子时,丁传给甲、乙、丙的概率分别为
,
,
.假设毽子一直没有掉地上,经过
次传毽子后,毽子被甲、乙、丙、丁接到的概率分别为
,
,
,
,已知
.
(1)记丁在前2次传毽子中,接到毽子的次数为
,求
的分布列;
(2)证明
为等比数列,并判断经过150次传毽子后甲接到毽子的概率与
的大小.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e6486784415f3537c9a13556c05d893.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e6486784415f3537c9a13556c05d893.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e6486784415f3537c9a13556c05d893.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.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/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c59e7c7a84a4bdb959e95536d0404ceb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82e260b088f071983f254ce8f5163fcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1bae03ee4ac75dacfb026290e4207dd.png)
(1)记丁在前2次传毽子中,接到毽子的次数为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(2)证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecbf26c8dca34c3a2742d1b5643469a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d266a04f3dc7483eddbc26c5e487db.png)
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名校
4 . 如图,有一列曲线
,
,……,
,……,且
1是边长为1的等边三角形,
是对
进行如下操作而得到:将曲线
的每条边进行三等分,以每边中间部分的线段为边,向外作等边三角形,再将中间部分的线段去掉得到
,记曲线
的边数为
,周长为
,围成的面积为
,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/deebd2c36a5e644a566f1980091359bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89e32039addb008103a2a8344225214a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e00d8f90655e6341907aa9c7c62d4398.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cffa35373ec4e4684107b42adb7a5161.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7a782bd6947ee3a8e0cf6d730ff4fd3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1efa3c897c73db3b2ad736035c6c961.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/452b9bcf720355d0678d62cbf6857ffe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7a782bd6947ee3a8e0cf6d730ff4fd3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97ea1a0db278d46806cf2a370f7bfcb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5a2d3cd8e283ae9d04bee5ab2e0895b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1cdf517b6ea59db5762a06830f23e47.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
A.数列{![]() |
B.数列{![]() ![]() |
C.数列![]() ![]() ![]() |
D.当n无限增大时,![]() ![]() |
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5卷引用:辽宁省沈阳市五校协作体2022-2023学年高二下学期期中考试数学试题
辽宁省沈阳市五校协作体2022-2023学年高二下学期期中考试数学试题湖南省常德市2023届高三下学期一模数学试题江西省上饶市民校考试联盟2022-2023学年高二下学期阶段测试(四)数学试题(已下线)专题6 等比数列的判断(证明)方法 微点2 通项公式法、前n项和公式法(已下线)模块四 专题1 期中重组篇(辽宁卷)(人教B版高二下学期)
名校
解题方法
5 . 记
为数列
的前
项和,已知
,且
.
(1)证明:
是等比数列;
(2)若
是等差数列,且
,
,求集合
中元素的个数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f16fe3ffe7d9e8246282ddb00fc5f4b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc9efeb4455e30293d412938eeea85d5.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43b7e7cd571c8cd141cbbfe5d0890bf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3aaee408bdec05bbdfcd4b841a331e61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c271ad58afa6d2a0a2a9215f81264c64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ee3f3ce4b539495cd57c650b6ec6a59.png)
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5卷引用:辽宁省朝阳市建平县2022-2023学年高三上学期9月月考数学试题
辽宁省朝阳市建平县2022-2023学年高三上学期9月月考数学试题河北省三河市2023届高三上学期开学联考数学试题(已下线)上海市华东师范大学第二附属中学2023届高三上学期10月月考数学试题云南省部分学校2023届高三上学期9月联考数学试题(已下线)专题17 数列(练习)-2
6 . 已知数列
满足
,
;设等差数列
、
的前
项和分别为
和
,且
,
,
.
(1)求证数列
是等比数列;
(2)求常数
的值及
的通项公式;
(3)求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50e1ee88beaddafb0d0a185c3a8e0dc5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b18c05bfbed0bec6f7e07c83edab8241.png)
![](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/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/750c58125af66fc90dd5e66119e92088.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1657d35123dad9edeaf674ce3e26970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85f7519e1b1dd927bc634eedafc88820.png)
(1)求证数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea399f670971aac0017799df7b344bdc.png)
(2)求常数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce1d227ff0547c85eb6b5d7d0a4628a7.png)
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名校
7 . 2020年春天随着疫情的有效控制,高三学生开始返校复课学习.为了减少学生就餐时的聚集排队时间,学校食堂从复课之日起,每天中午都会提供
、
两种套餐(每人每次只能选择其中一种),经过统计分析发现:学生第一天选择
类套餐的概率为
、选择
类套餐的概率为
.而前一天选择了
类套餐第二天选择
类套餐的概率为
、选择
套餐的概率为
;前一天选择
类套餐第二天选择
类套餐的概率为
、选择
类套餐的概率也是
,如此往复.记某同学第
天选择
类套餐的概率为
.
(1)证明数列
是等比数列,并求数列
的通项公式;
(2)记高三某宿舍的3名同学在复课第二天选择
类套餐的人数为
,求
的分布列并求
;
(3)为了贯彻五育并举的教育方针,培养学生的劳动意识,一个月后学校组织学生利用课余时间参加志愿者服务活动,其中有20位学生负责为全体同学分发套餐.如果你是组长,如何安排分发
、
套餐的同学的人数呢,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf31876698721a199c7c53c6b320aa86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d266a04f3dc7483eddbc26c5e487db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b2a698891d42c70b597f0da4f215f09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf83e20035c3afd6d26ebfd53d768a70.png)
(1)证明数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2ddf17be3250288052d71b2cb3d24b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c5a325806df1a1c3e7ce609fe99085f.png)
(2)记高三某宿舍的3名同学在复课第二天选择
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bf3baba074e8aeb6f3ea117865bbd1b.png)
(3)为了贯彻五育并举的教育方针,培养学生的劳动意识,一个月后学校组织学生利用课余时间参加志愿者服务活动,其中有20位学生负责为全体同学分发套餐.如果你是组长,如何安排分发
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
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6卷引用:辽宁省葫芦岛市四校2022-2023学年高三上学期期中数学试题
辽宁省葫芦岛市四校2022-2023学年高三上学期期中数学试题山东省实验中学2021届高三下学期一模数学试题(已下线)重难点04 概率与统计-2022年高考数学【热点·重点·难点】专练(新高考专用)(已下线)押全国卷(理科)第18题 概率与统计-备战2022年高考数学(理)临考题号押题(全国卷)(已下线)专题1 “五育并举”类型(已下线)2024届新高考数学原创卷4