23-24高二下·全国·课前预习
1 . 等比数列前
项和公式的函数特征
(1)当公比
时,设
,等比数列的前
项和公式是
,即
是
的________ (2)当公比
时,因为
,所以
是
的________ .
温馨提醒:当
,所以
的结构形式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(1)当公比
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45482d31d1d7448c9f3922b4d2a55331.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9eb685890a92dfb9abf497f6a28d1c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d7eacf0e45686808894857edcfddcd6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfa9bf65189dfb57a61644a1cb27f361.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc9efeb4455e30293d412938eeea85d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9285e2558a49fd5b378dde4878ac346.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
温馨提醒:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c81333addf3c81ac7ea2341e513d5308.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c94cdb9f5e337d904ed26230e2406291.png)
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23-24高二下·全国·课前预习
2 . 等比数列的前
项和公式
注:用等比数列前
项和公式求和,一定要对该数列的公比________ ,进行分类讨论;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
已知量 | 首项、公比和项数 | 首项、末项和公比 |
公式 | ![]() | ![]() |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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23-24高二下·全国·课前预习
3 . 数学归纳法的操作流程
(1)________ 奠基要稳,有些问题中验证的初始值
不一定为1.
(2)正确分析由
到
时式子________ 是应用数学归纳法成功证明问题的保障.
(3)在第二步证明中一定要________ ,这是数学归纳法证明的核心环节,否则这样的证明就不是利用数学归纳法证明.
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16530bfffc3b0bb4bda872bf43a3b82f.png)
(2)正确分析由
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2b66d04abdc608824821dee4c842065.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e8d70d8c5c609c5b55dd2d795be9648.png)
(3)在第二步证明中一定要
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4 . 裂项求和
把数列的通项拆成两项之差,在求和时中间的一些项可以相互抵消,从而求得前
项和.
裂项时常用的五种变形:
(1)![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac067a9b66a44acd5abaa0c021b5e3e5.png)
______
;
(2)![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d06787f1848280ca6ca23d14da193321.png)
______ .
(3)![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4efaa1a53624d5942626cf9bddc288d3.png)
______
;
(4)![](https://staticzujuan.xkw.com/quesimg/Upload/formula/235c44b3845afe108250918d8b41a339.png)
______ .
(5)若数列
是等差数列,且公差
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0ade8d55a7d63674831f8c559ab5707.png)
______ .
把数列的通项拆成两项之差,在求和时中间的一些项可以相互抵消,从而求得前
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
裂项时常用的五种变形:
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac067a9b66a44acd5abaa0c021b5e3e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3cb77304ba3546c1fc629832f1b811f.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d06787f1848280ca6ca23d14da193321.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4efaa1a53624d5942626cf9bddc288d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3cb77304ba3546c1fc629832f1b811f.png)
(4)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/235c44b3845afe108250918d8b41a339.png)
(5)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/812be9806122241c476ba1db516c4823.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0ade8d55a7d63674831f8c559ab5707.png)
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23-24高二下·全国·课前预习
解题方法
5 . 等差数列前
项和的性质
(1)若数列
是公差为
的等差数列,则数列
也是等差数列,且公差为______ .
(2)若
分别为等差数列
的前
项,前
项,前
项的和,则
,
也成等差数列,公差为______ .
(3)设两个等差数列
的前
项和分别为
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6322378e9dc138599481f035cfe3b38.png)
______ .
(4)在等差数列中,若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be2e0d4d2e550f00b36d6f00111418ba.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(1)若数列
![](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/4dea1dd4ffcb4cf0697ca43079f6a1f2.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35ddff98f658432f3723f43951abd46e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fad491e5b5e14c49ef8b7004ebcfcef9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ade9841a8e6840efddcfd8620a6fc1fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b477afc102fe376cc777fffe0548cb0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aacaf4a1b543085ebf2617cd600c011a.png)
(3)设两个等差数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62e567d7e9761951a266953c8d5042ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/478421b81927e435cbcf5acafa89efd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6322378e9dc138599481f035cfe3b38.png)
(4)在等差数列中,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bcbc6ca5f2b222970ce2473603d54b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be2e0d4d2e550f00b36d6f00111418ba.png)
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名校
6 . 已知抛物线
的焦点为
,过
在第一象限上的任意一点
作
的切线
,直线
交
轴于点
.过
作
的垂线
,交
于
两点.
(1)若点
在
的准线上,求直线
的方程;
(2)求
的中点
的轨迹方程;
(3)若三角形
面积为
,求点
的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c549074bfdfdc639af9880193c891e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
(1)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fb26d84907c923278ac4626a9d58947.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(3)若三角形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
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名校
7 . “踩高跷,猜灯谜”是我国元宵节传统的文化活动. 某地为了弘扬文化传统,发展“地摊经济”,在元宵节举办形式多样的猜灯谜活动.
(1)某商户借“灯谜”活动促销,将灯谜按难易度分为
两类,抽到较易的
类并答对购物打八折优惠,抽到稍难的
类并答对购物打七折优惠,抽取灯谜规则如下:在一不透明的纸箱中有8张完全相同的卡片,其中3张写有
字母,3张写有
字母,2张写有
字母,顾客每次不放回从箱中随机取出1张卡片,若抽到写有
的卡片,则再抽1次,直至取到写有
或
卡片为止,求该顾客取到写有
卡片的概率.
(2)小明尝试去找全街最适合他的灯谜,规定只能取一次,并且只可以向前走,不能回头,他在街道上一共会遇到
条灯谜(不妨设每条灯谜的适合度各不相同),最适合的灯谜出现在各个位置上的概率相等,小明准备采用如下策略:不摘前
条灯谜,自第
条开始,只要发现比他前面见过的灯谜适合的,就摘这条灯谜,否则就摘最后一条,设
,记小明摘到那条最适合的灯谜的概率为
.
①若
,
,求
;
②当
趋向于无穷大时,从理论的角度,求
的最大值及
取最大值时
的值.(取
)
(1)某商户借“灯谜”活动促销,将灯谜按难易度分为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da6b93dbe5272a5167ff4e2918bec864.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](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/c5db41a1f31d6baee7c69990811edb9f.png)
![](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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
(2)小明尝试去找全街最适合他的灯谜,规定只能取一次,并且只可以向前走,不能回头,他在街道上一共会遇到
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/578606f2403c782b320a303b139dc626.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b00f4eb7f1bd2ccefbabf0c1dfa8f69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dd266684a38a50f7a7925d4bb5e63e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fac3649308b528fd56545ba102dc42d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8e69866076dcff686a05e9e91e61e68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
②当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a5a5ce3e0a6cbb682718706600b239c.png)
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2024-04-22更新
|
1295次组卷
|
3卷引用:专题07 概率与统计综合问题(6类题型)-备战2023-2024学年高二数学下学期期末真题分类汇编(江苏专用)
(已下线)专题07 概率与统计综合问题(6类题型)-备战2023-2024学年高二数学下学期期末真题分类汇编(江苏专用)江苏省海安高级中学、宿迁中学2023-2024学年高三下学期模拟考试数学试卷黑龙江省大庆市实验中学实验二部2024届高三下学期得分训练数学试题(六)
23-24高二下·全国·课前预习
8 . 数列的前
项和![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
数列前
项和的概念数列
从第1项起到第
项止的各项之和,称为数列
的前
项和,记作
,即
______ .
数列的前
项和
与通项
的关系若数列
的前
项和为
,则当
时,
;当
时,
;则
________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
数列前
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](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/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/04c2864e2ec3416cc4c081ac1f71a0af.png)
数列的前
![](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/96abfe2da27a63e6affb19a0c80236d9.png)
![](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/c87b351f16728b0023fd63678f8103c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd8d9e0c9781bac9ba899c3c5a7bdb9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cec7b88f0fe2981434a30e1fd72b782e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3f7fda69e2b32b9ced2239f915fa59b.png)
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23-24高二下·全国·课前预习
9 . 数列与函数的关系
从函数的观点看,数列可以看作是特殊的函数,关系如下表:
从函数的观点看,数列可以看作是特殊的函数,关系如下表:
定义域 | ![]() |
解析式 | 数列的通项公式 |
值域 | 自变量从1开始,按照 |
表示方法 | (1)通项公式(解析法);(2) |
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名校
解题方法
10 . 某学校工会组织趣味投篮比赛,每名选手只能在下列两种比赛方式中选择一种.
方式一:选手投篮3次,每次投中可得1分,未投中不得分,累计得分;
方式二:选手最多投3次.如第1次投中可进行第2次投篮,如第2次投中可进行第3次投篮.如某次未投中,则投篮中止.每投中1次可得2分,未投中不得分,累计得分;
已知甲选择方式一参加比赛,乙选择方式二参加比赛.假设甲,乙每次投中的概率均为
,且每次投篮相互独立.
(1)求甲得分不低于2分的概率;
(2)求乙得分的分布列及期望;
(3)甲,乙谁胜出的可能性更大?直接写出结论.
方式一:选手投篮3次,每次投中可得1分,未投中不得分,累计得分;
方式二:选手最多投3次.如第1次投中可进行第2次投篮,如第2次投中可进行第3次投篮.如某次未投中,则投篮中止.每投中1次可得2分,未投中不得分,累计得分;
已知甲选择方式一参加比赛,乙选择方式二参加比赛.假设甲,乙每次投中的概率均为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
(1)求甲得分不低于2分的概率;
(2)求乙得分的分布列及期望;
(3)甲,乙谁胜出的可能性更大?直接写出结论.
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