解题方法
1 . 若实数列
满足
,有
,称数列
为“
数列”.
(1)判断
是否为“
数列”,并说明理由;
(2)若数列
为“
数列”,证明:对于任意正整数
,且
,都有![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce97bb4c969108ebef4ebadd5acc5ca4.png)
(3)已知数列
为“
数列”,且
.令
,其中
表示
中的较大者.证明:
,都有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08fee3639b8e3ff9717f3ffd9633927f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8414472e2121e1796eb40408d820053a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
(1)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81f1bc95528c774ce919f7c5a0ef0d5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4d7e68d0c8bd9a32d826c721ab74d9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22e517c32ed93de4d9cb6e0926337000.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce97bb4c969108ebef4ebadd5acc5ca4.png)
(3)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93846b72089bd425864969f2edabdb8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8957564d30af14db69fdc36be2eaee7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/065f8a641aab5c39f73be89a6f2e7a65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4dc7e8bd20e9d38b4d248a8c253ccd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86e8131eb05d480f0da6c61ebda3ce48.png)
您最近一年使用:0次
2 . 某校高一学生1000人,每周一次同时在两个可容纳600人的会议室,开设“音乐欣赏”与“美术鉴赏”的校本课程.要求每个学生都参加,要求第一次听“音乐欣赏”课的人数为
,其余的人听“美术鉴赏”课;从第二次起,学生可从两个课中自由选择.据往届经验,凡是这一次选择“音乐欣赏”的学生,下一次会有20%改选“美术鉴赏”,而选“美术鉴赏”的学生,下次会有30%改选“音乐欣赏”,用
,
分别表示在第
次选“音乐欣赏”课的人数和选“美术鉴赏”课的人数.
(1)若
,分别求出第二次,第三次选“音乐欣赏”课的人数
,
;
(2)①证明数列
是等比数列,并用n表示
;
②若要求前十次参加“音乐欣赏”课的学生的总人次不超过5800,求m的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a4643842b22bc7d26e43000111359e3.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/b6a24198bd04c29321ae5dc5a28fe421.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c06b1a798196b196c70d42f9a5b40b43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
(2)①证明数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cb61e05a3be8310c15cda0ab0fc91b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
②若要求前十次参加“音乐欣赏”课的学生的总人次不超过5800,求m的取值范围.
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3 . 在通信技术中由
和
组成的序列有着重要作用,序列中数的个数称为这个
序列的长度
如
是一个长度为
的
序列
长为
的
序列中任何两个
不相邻的序列个数设为
,长度为
的
序列为:
,
,都满足数列
,
长度为
且满足数列
的
序列为:
,
,
,
.
(1)求
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/634e70033857b751723d34d1ca86f375.png)
(2)求数列
中
,
,
的递推关系![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32ca6fa9955690cec01db601e3abce0c.png)
(3)记
是数列
的前
项和,证明:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c95b6be4554f03bf496092f1acdfbb89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ffaa8e2bd299bb83168cfac17137d29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c90282d4a37c9a20620d4bbb0c263cae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aec18fcd4df134d5037dd56f3d82841.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f43677db00ba65a7f96fc49627d70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ffaa8e2bd299bb83168cfac17137d29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c90282d4a37c9a20620d4bbb0c263cae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ffaa8e2bd299bb83168cfac17137d29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bfccafa83afe5ee21eab6ef2b2c8852.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ffaa8e2bd299bb83168cfac17137d29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c95b6be4554f03bf496092f1acdfbb89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe229b24e2d56ff6b491725ceae4ff2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2581813c187d2e230d97567d649d72b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe229b24e2d56ff6b491725ceae4ff2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ffaa8e2bd299bb83168cfac17137d29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e222c99263ff290929466f52bdb07404.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01a3257d015e9b178850734cfc3a5b1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f43677db00ba65a7f96fc49627d70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d786bde127368fd1b2ed7a7d233db866.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cbeed5324c432101be517dd6f5c735b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/634e70033857b751723d34d1ca86f375.png)
(2)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe229b24e2d56ff6b491725ceae4ff2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d41ae64e37ebcddccabd64e12b0afc2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7085d141e33ba0188e58fa2177d89ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bfccafa83afe5ee21eab6ef2b2c8852.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32ca6fa9955690cec01db601e3abce0c.png)
(3)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b708c8dcb2d66eb2ce0b3718a9cd924a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe229b24e2d56ff6b491725ceae4ff2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b1c13936c1d87bc8fc19a215f8a138f.png)
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名校
4 . 某运动员多次对目标进行射击, 他第一次射击击中目标的概率为
.由于受心理因素的影响,每次击中目标的概率会受前一次是否击中目标而改变,若前一次击中目标,下一次击中目标的概率为
;若前一次末击中目标,则下一次击中目标的概率为
.
(1)记该运动员第
次击中目标的概率为
,证明:
为等比数列,并求出
的通项公式;
(2)若该运动员每击中一次得2分,未击中不得分,总共射击2次,求他总得分
的分布列与数学期望.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eac97e6740365c85ad857aff85cefbe5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b2a698891d42c70b597f0da4f215f09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
(1)记该运动员第
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf83e20035c3afd6d26ebfd53d768a70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29a0fd459ceae1ed8a031ab462546d09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c5a325806df1a1c3e7ce609fe99085f.png)
(2)若该运动员每击中一次得2分,未击中不得分,总共射击2次,求他总得分
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
您最近一年使用:0次
2022-09-23更新
|
1312次组卷
|
4卷引用:云南师范大学附属中学2023届高三上学期适应性月考卷(三)数学试题
5 . 阿司匹林(分子式
,分子质量180)对血小板聚集的抑制作用,使它能降低急性心肌梗死疑似患者的发病风险.对于急性心肌梗死疑似患者,建议第一次服用剂量300
,嚼碎后服用以快速吸收,以后每24小时服用200
.阿司匹林口服后经胃肠道完全吸收,阿司匹林吸收后迅速降解为主要代谢产物水杨酸(分子式
,分子质量138),降解过程生成的水杨酸的质量为阿司匹林质量的
,水杨酸的清除半衰期(一般用物质质量衰减一半所用的时间来描述衰减情况,这个时间被称作半衰期)约为12小时.(考虑所有阿司匹林都降解为水杨酸)
(1)求急性心肌梗死疑似患者第1次服药48小时后第3次服药前血液中水杨酸的含量(单位
);
(2)证明:急性心肌梗死疑似患者服药期间血液中水杨酸的含量不会超过230
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac7bf7ba2db0fd1143b1d51e381fd6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/260c686e89023f3f7d2879e70fca0605.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/260c686e89023f3f7d2879e70fca0605.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ac54805982e3b5d68309b106be01176.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5744217027ca031a07632d9678d1f213.png)
(1)求急性心肌梗死疑似患者第1次服药48小时后第3次服药前血液中水杨酸的含量(单位
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/260c686e89023f3f7d2879e70fca0605.png)
(2)证明:急性心肌梗死疑似患者服药期间血液中水杨酸的含量不会超过230
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/260c686e89023f3f7d2879e70fca0605.png)
您最近一年使用:0次
2022-05-02更新
|
347次组卷
|
3卷引用:浙江省杭州地区(含周边重点中学)2021-2022学年高二下学期期中联考数学试题
浙江省杭州地区(含周边重点中学)2021-2022学年高二下学期期中联考数学试题重庆市五校2022届高三上学期10月联考数学试题(已下线)专题09 数列的通项公式、数列求和及综合应用(9大核心考点)(讲义)
名校
6 . 已知某同学在任何一次拓展考试中获得满分的概率都为
,且各次考试的成绩相互独立.以
表示他参加n(
,
)次考试后从未连续取得2次满分的概率.
(1)求
,
的值,并证明当n≥4时,
;
(2)证明:对任意
,
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf83e20035c3afd6d26ebfd53d768a70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b9cb8e6ff801523b0304576cd69fd2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/797e67927616b141ed7c6b83f8b6f4fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6ee4ba94b630d7a4015633ce7556d35.png)
(2)证明:对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a787f7a6b80d21a2c98e1c65fca8de94.png)
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7 . 设数列
,
,
,…满足条件
,
其中n是某个固定的自然数.求证:
.
![](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/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3954c19f9f2b479d44a6b2c910343d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92d5d2e376e3cbae1057d3c34bb7dfba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9baf2da944bd8917bb14235f0c350f85.png)
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8 . 治理垃圾是
地改善环境的重要举措.去年
地产生的垃圾量为200万吨,通过扩大宣传、环保处理等一系列措施,预计从今年开始,连续5年,每年的垃圾排放量比上一年减少20万吨,从第6年开始,每年的垃圾排放量为上一年的
.
(1)写出
地的年垃圾排放量与治理年数
的表达式;
(2)设
为从今年开始
年内的年平均 垃圾排放量,证明数列
为递减数列;
(3)通过至少 几年的治理,
地的年平均垃圾排放量能够低于100万吨?
![](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/d6e2cd5f05dfeea38c37dc88669b830d.png)
(1)写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ea53f8c223d61d151ce2ab8c18ca374.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aadd51d72723320ae712a8a7622551cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97ee58fbaca68cb593091abd35e9cb6c.png)
(3)通过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
您最近一年使用:0次
2022-01-15更新
|
412次组卷
|
2卷引用:北京市大兴区2021-2022学年高二上学期期末检测数学试题
名校
9 . 冠状病毒是一个大型病毒家族,已知可引起感冒以及中东呼吸综合征(MERS)和严重急性呼吸综合征(SARS)等较严重疾病.新型冠状病毒是以前从未在人体中发现的冠状病毒新毒株,人感染了冠状病毒后常见体征有呼吸道症状、发热、咳嗽、气促和呼吸困难等.日前正在世界范围内广泛传播,并对人类生命构成了巨大威胁.针对病毒对人类的危害,科研人员正在不断研发冠状病毒的抑制剂.某种病毒抑制剂的有效率为60%,现设计针对此抑制剂的疗效试验:每次对病毒使用此抑制剂,如病毒被抑制,得分为2分,如抑制剂无效,得分1分,持续进行试验.设得分为
时的概率为
.
(1)进行两次试验后,总得分为随机变量
,求
的分布列和数学期望;
(2)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df37d45d84dcc1fa4d42cf209b935eea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf83e20035c3afd6d26ebfd53d768a70.png)
(1)进行两次试验后,总得分为随机变量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e861c7f72f74db0a67e6b272ba2aae5a.png)
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