1 . 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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名校
2 . 若数列
满足
,从数列
中任取2项相加,把所有和的不同值按照从小到大排成一列,称为数列
的和数列,记作数列
.
(1)已知等差数列
的前n项和为
,且
.
①若
,
,求
的通项公式,并写出
的前5项;
②若
,
,求数列
的前50项的和;
(2)若
,证明:对任意
或
,
,并求数列
的所有项的和.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1a5945ce5c2114af8c18718ca8dc899.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62c320a0619c63a5b650a1a94c0a5679.png)
(1)已知等差数列
![](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/a1a5945ce5c2114af8c18718ca8dc899.png)
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58365ff21052f2f978c11844b002b933.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb3fdeeb4afe6485ffb00bf83023e704.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62c320a0619c63a5b650a1a94c0a5679.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/751859e4f0b1cb2c94fd5cca373de9af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a50c3a2b8abc17a7e110f9811296a05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62c320a0619c63a5b650a1a94c0a5679.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/559497cb5b10c9c489ee0cdc11fa2a7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12329f3ac81209a815f8c4fa12c4b6cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d149f4ed2b72f3e3ee850e163ba35473.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e23ba0aeb43a20799d1f414650203ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62c320a0619c63a5b650a1a94c0a5679.png)
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2024-05-06更新
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2卷引用:江西省抚州市金溪县第一中学等校2023-2024学年高二下学期期中考试数学试卷
3 . 已知数列
的前
项和为
,若数列
满足:①数列
项数有限为
;②
;③
,则称数列
为“
阶可控摇摆数列”.
(1)若等比数列
为“10阶可控摇摆数列”,求
的通项公式;
(2)若等差数列
为“
阶可控摇摆数列”,且
,求数列
的通项公式;
(3)已知数列
为“
阶可控摇摆数列”,且存在
,使得
,探究:数列
能否为“
阶可控摇摆数列”,若能,请给出证明过程;若不能,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.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/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ccd4ed75729a7f7a2d5a3d9f7293c53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1798fb0c31c65218cd20e07320a17d86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
(1)若等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bdaa641d2e7e17904c61ff7245a5cb0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
(2)若等差数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e7364bbda64feeb4d448f9316d4c67a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fad491e5b5e14c49ef8b7004ebcfcef9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa22ba45c62adc96ffe508594edd6900.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
(3)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daca8076f0553088afded57b48009d37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ae2ea9de54e074c145b8259f6c55e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d013861990cf331c82eb453416ae31bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
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6卷引用:江西省2024届高三下学期二轮复习阶段性检测数学试题
江西省2024届高三下学期二轮复习阶段性检测数学试题吉林省白山市2024届高三第二次模拟考试数学试题山东省淄博市实验中学2023-2024学年高二下学期第一次月考(3月)数学试卷(已下线)数学(广东专用01,新题型结构)吉林省通化市梅河口市第五中学2024届高三下学期二模数学试题(已下线)压轴题05数列压轴题15题型汇总-1
解题方法
4 . 对于数列
,若满足
恒成立的最大正数
为
,则称
为“
数列”.
(1)已知等比数列
的首项为1,公比为
,且为“
数列”,求
;
(2)已知等差数列
与其前
项和
均为“
数列”,且
与
的单调性一致,求
的通项公式;
(3)已知数列
满足
,若
且
,证明:存在实数
,使得
是“
数列”,并求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fec1e634e69670226b7aa4af264b9f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ed1e9cdd5a82f29ec89b2c53b4fa6f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7240c65faee0ddb7b65aaaf02f5790e.png)
(1)已知等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/407cd2b4e2b6f2d503662200da4c84fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
(2)已知等差数列
![](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/846fa57d92d6ad44d6a0cafad1e71ed4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7240c65faee0ddb7b65aaaf02f5790e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/846fa57d92d6ad44d6a0cafad1e71ed4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f44ef293afdf4ba3dfa03f580e71f5dc.png)
(3)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/115d32d0cc00727c43c1f27ed846c805.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/699e55c211a6e091cc7a9d2cde3ed981.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07cfaba0cbf55e596e66953eb795f757.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ed1e9cdd5a82f29ec89b2c53b4fa6f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2b58298b057a73ed8e2dde655161046.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7240c65faee0ddb7b65aaaf02f5790e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ed1e9cdd5a82f29ec89b2c53b4fa6f8.png)
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2024-03-21更新
|
136次组卷
|
2卷引用:江西省部分学校2023-2024学年高二下学期第一次阶段性考试数学试卷
名校
5 . 公元263年,刘徽首创了用圆的内接正多边形的面积来逼近圆面积的方法,算得
值为3.14,我国称这种方法为割圆术,直到1200年后,西方人才找到了类似的方法,后人为纪念刘徽的贡献,将3.14称为徽率.我们作单位圆的外切和内接正
边形
,记外切正
边形周长的一半为
,内接正
边形周长的一半为
.通过计算容易得到:
(其中
是正
边形的一条边所对圆心角的一半)
(1)求
的通项公式;
(2)求证:对于任意正整数
依次成等差数列;
(3)试问对任意正整数
是否能构成等比数列?说明你的理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbbc0cf9164007ddd298dd2236703f2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bbccb799ae7eb992b25b2426173ed36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbbc0cf9164007ddd298dd2236703f2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbbc0cf9164007ddd298dd2236703f2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96936fc2a366e6a8d1dfae54322d5d4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92ffa8be5a02790c6161c56b8e90db64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbbc0cf9164007ddd298dd2236703f2f.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)求证:对于任意正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ac64c640ccd57708681eada27a8fa6d.png)
(3)试问对任意正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8e42bf4d8449d427c1f5f252db0f298.png)
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2023-07-21更新
|
382次组卷
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3卷引用:江西省宜春市丰城中学2023-2024学年高二下学期4月期中考试数学试题
名校
解题方法
6 . 某排球教练带领甲、乙两名排球主力运动员训练排球的接球与传球,首先由教练第一次传球给甲、乙中的某位运动员,然后该运动员再传回教练.每次教练接球后按下列规律传球:若教练上一次是传给某运动员,则这次有的概率再传给该运动员,有
的概率传给另一位运动员.已知教练第一次传给了甲运动员,且教练第
次传球传给甲运动员的概率为
.
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adad9633b73dfbbb3d84b4f15979e99e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87b21b872313f7d8c5b606981f954a1e.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ffb021aa7d5a5c2f0691e337caad624.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e4f333edad039c5c879218b5c815cae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a133513458bd3ecec0b759575cdc6b82.png)
您最近一年使用:0次
2023-12-05更新
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1868次组卷
|
6卷引用:江西省上饶市广丰中学2023-2024学年高一上学期12月月考数学试题
江西省上饶市广丰中学2023-2024学年高一上学期12月月考数学试题河北省部分重点高中2024届高三高考模拟数学试题(已下线)模块二 专题5 概率中的创新问题(已下线)黄金卷04(已下线)专题8-2分布列综合归类-2(已下线)【一题多变】传球问题 构造数列
7 . 已知点
,
,设
,当
时,线段
的中点为
,
关于直线
的对称点为
.例如,
为线段
的中点,则
,
.
(1)设
,证明:
是等比数列.
(2)求数列
的通项公式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89f852312e160b9bb57b86135bd88789.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b23593cb29fb4b4fdcb8fb95583c4c23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53fcfca2a223425da57d1f24c98640dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69a4f2bafc3346d14a8967ce2546157d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63f5c583c98a1fd516c6ceaa60b55dec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63f5c583c98a1fd516c6ceaa60b55dec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d77f5191798242b7b9b88a75e17e4425.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3cfeacc29e6a61c5b3b4e439c0a91df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2857fac4963b129d99e79dcb3e13d295.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/473913c0887bb64d386f4c02f1853452.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6956cd853698a7c6c54fc2d6173d8f3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb428a0eceb551ea36598753fef1f995.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65028e4ccc71adf10353c77b71cf4aa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
(2)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5344eadd4711db34e3f935aedd5fb270.png)
您最近一年使用:0次
2023-12-22更新
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725次组卷
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7卷引用:江西省“三新”协同教研共同体2024届高三上学期12月联考数学试题
(已下线)江西省“三新”协同教研共同体2024届高三上学期12月联考数学试题江西省部分学校2024届高三上学期12月联考数学试题江西省赣州市南康中学2024届高三上学期新高考“七省联考”考前数学猜题卷(一)河北省邢台市部分重点高中2023-2024学年高二上学期1月期末数学试题(已下线)考点4 等比数列的定义与判断 2024届高考数学考点总动员(已下线)考点9 数列通项公式 2024届高考数学考点总动员【练】(已下线)第5讲:数列模型的应用【练】
2023高三·全国·专题练习
8 . 已知数列
的前n项和是
,且
.
(1)证明:
为等比数列;
(2)证明:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89a5bcdb7e6d6818254e00d02645e271.png)
(3)
为数列
的前n项和,设
,是否存在正整数m,k,使
成立,若存在,求出m,k;若不存在,说明理由.
![](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/e3c07d6d0c63061e09e36b5a2c74760b.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e2de706dc5f0439b989273a5367f63a.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89a5bcdb7e6d6818254e00d02645e271.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/878b4bc8b23c9f486874016f32221333.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a59fe8864b6c25590de9d542f54b9422.png)
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9 . 2021年4月23日是第26个“世界读书日”,某校组织“阅百年历程,传精神力量”主题知识竞赛,有基础题、挑战题两类问题.每位参赛同学回答
次
,每次回答一个问题,若回答正确,则下一个问题从挑战题库中随机抽取;若回答错误,则下一个问题从基础题库中随机抽取.规定每位参赛同学回答的第一个问题从基础题库中抽取,基础题答对一个得10分,否则得0分;挑战题答对一个得30分,否则得0分.已知小明能正确回答基础类问题的概率为
,能正确回答挑战类问题的概率为
,且每次回答问题是相互独立的.
(1)记小明前2题累计得分为
,求
的概率分布列和数学期望;
(2)记第
题小明回答正确的概率为
,证明:当
时,
,并求
的通项公式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94915db9979f1a450f8e1cb1a05dc65d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eac97e6740365c85ad857aff85cefbe5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d33adb74906403b0b00fcbd9fa691d8b.png)
(1)记小明前2题累计得分为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(2)记第
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eda87cf561371849b59044a74199dcda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c972cbd63decec197aec1bdc306de67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf92644e054a9ba9ad3ddcf05cc3bb3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad4546b288340a9393260ed532171518.png)
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2022-02-08更新
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7卷引用:江西省智学联盟体(新余市第一中学、南康中学等)2022-2023学年高二第二次联考数学试题
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