1 . 设数列
的前
项和为
,且
与
的等差中项为
.
(1)证明:数列
是等比数列;
(2)设
,证明:
.
![](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/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47b6f30d133e86188d9e5185720bdbb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6ac430c4ce342d868e9399ef45ed4d4.png)
(1)证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a080c94bf1ffea8d5af10f9688978fb5.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2414b9e4bf818edab0bfb752bc73aeb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6b6aa1778c3b6fd9d5e148da3f34f1e.png)
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2 . 把正整数按下图所示的规律排序,则从2021到2023的箭头方向依次为( )
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/14/f14b672b-55af-4b4e-9aa6-889600e44a78.png?resizew=262)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/14/f14b672b-55af-4b4e-9aa6-889600e44a78.png?resizew=262)
A.![]() | B.![]() |
C.![]() | D.![]() |
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3 . 容器中有![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
种粒子,若相同种类的两颗粒子发生碰撞,则变成一颗B粒子;不同种类的两颗粒子发生碰撞,会变成另外一种粒子.例如,一颗A粒子和一颗B粒子发生碰撞则变成一颗C粒子,现有A粒子10颗,B粒子8颗,C粒子9颗,如果经过各种两两碰撞后,只剩1颗粒子.给出下列结论:
①最后一颗粒子可能是A粒子;
②最后一颗粒子可能是B粒子;
③最后一颗粒子可能是C粒子;
其中正确结论的序号是______ .(写出所有正确结论的序号)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
①最后一颗粒子可能是A粒子;
②最后一颗粒子可能是B粒子;
③最后一颗粒子可能是C粒子;
其中正确结论的序号是
您最近一年使用:0次
2023-02-08更新
|
764次组卷
|
4卷引用:2023年四省联考变试题11-16
名校
4 . 已知
,
.
(1)求
在
处的切线方程;
(2)求证:对于
和
,且
,都有
;
(3)请将(2)中的命题推广到一般形式,井用数学归纳法证明你所推广的命题.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa7f9b35017daa8b524c5717a355834a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd5cdde751120c6deab563a6f7f8cf05.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17c3319647314c3b6d82958a909acd2a.png)
(2)求证:对于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65fd2742daefe770eca5c2270b504f9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3f97f4caf938dc3b05889a363ab8ee0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a85ea4968343b0d94ed2fe01b535.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b23755a25b5bf295b3533dc94f70651f.png)
(3)请将(2)中的命题推广到一般形式,井用数学归纳法证明你所推广的命题.
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5 .
名学生参加某次测试,测试由
道题组成.若一道题至少有
名学生未解出来,则称此题为难题;若一名学生至少解出了
道题,则该生本次测试成绩合格.如果这次测试至少有
名学生成绩合格,且测试中至少有
道题为难题,那么
的最小值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a75b0d2d4c86d3208ea797cc5a03ad7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0158fc47568ad3116bae022d129d9fe7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a75b0d2d4c86d3208ea797cc5a03ad7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0158fc47568ad3116bae022d129d9fe7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1019d4ad2e3fb4a7abb66e0e9e55b556.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
您最近一年使用:0次
2023-03-27更新
|
747次组卷
|
3卷引用:北京市西城区2023届高三一模数学试题
6 . 2023年2月22日,中国厦门市一名8岁男孩用时4.305秒单手完成4层汉诺塔游戏,成为新的世界纪录保持者.汉诺塔游戏源于1883年法国数学家卢卡斯提出的汉诺塔问题,有
,
,
三根柱子,在
柱上放着由下向上逐渐变小的
个盘子,现要求把
柱上的盘子全部移到
柱上,且需遵循以下的移动规则:
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/1/d4751208-a68e-4736-a4fb-05fb20c3b7d7.png?resizew=191)
①每次只能移动一个盘子;
②任何时候都不允许大盘子放在小盘子的上面;
③移动过程中盘子可以放在
,
,
中任意一个柱子上.
若用
表示
个盘子时最小的移动次数,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98396740a39a6a2e1fbaf5f0b1978ea9.png)
______ ,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02cb4ec5b02c810fd3bb29f99d90d919.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/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/1/d4751208-a68e-4736-a4fb-05fb20c3b7d7.png?resizew=191)
①每次只能移动一个盘子;
②任何时候都不允许大盘子放在小盘子的上面;
③移动过程中盘子可以放在
![](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)
若用
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f2bf3a57d272f43de9f9108841183ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98396740a39a6a2e1fbaf5f0b1978ea9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02cb4ec5b02c810fd3bb29f99d90d919.png)
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2023高三·全国·专题练习
7 . 已知数列
满足:
且
.
(1)求证:
;
(2)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c9f07174a0fb6528127c054d2c018ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039e4fe671d61e59b96ee525c9df43e8.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/faad572dcf773cecd015bddd7e7206f0.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1a5945ce5c2114af8c18718ca8dc899.png)
您最近一年使用:0次
名校
8 . 对于定义在
上的函数
如果同时满足以下三个条件:①
;②对任意
成立;③当
时,总有
成立.则称
为“理想函数”.有下列两个命题:
命题
:若
为“理想函数”,则对任意
,都有
;
命题
:若
为“理想函数”,则对任意
,都有
成立.
则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e11f4ca0e7ace69f92130d0525bcdb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/249a976e88133f3b3733f09137cf5c42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e41e188f97515c589454c51fb8e751b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27d5ef9e6f5429c22535001e95d726d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9f6988640a0b9bc4f9637132e6ca470.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
命题
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3b39bbfd4894f4d2ca18473a3e42f82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61ee7abd882ba99660bca68ebf544cd6.png)
命题
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f7dbb416ec1ff1984a724a4f48bf692.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09d3c756a52c9185ae6d02d9b9312f29.png)
则下列说法正确的是( )
A.命题![]() ![]() |
B.命题![]() ![]() |
C.命题![]() ![]() |
D.命题![]() ![]() |
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9 . 在第24届北京冬奥会开幕式上,一朵朵六角雪花飘拂在国家体育场上空,畅想着“一起向未来”的美好愿景.如图是“雪花曲线”的一种形成过程:从一个正三角形开始,把每条边分成三等份,然后以各边的中间一段为底边分别向外作正三角形,再去掉底边,重复进行这一过程.若第1个图中的三角形的周长为1,记第n个图形的周长为
,
为数列
的前n项和,则
( )
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/11/6df7d01d-f096-4908-a54a-cb0aee725f2b.png?resizew=371)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0496f142d8ae5acb06e83526eaa3ef87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04c2864e2ec3416cc4c081ac1f71a0af.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/11/6df7d01d-f096-4908-a54a-cb0aee725f2b.png?resizew=371)
A.![]() | B.![]() |
C.![]() | D.![]() |
您最近一年使用:0次
2023-02-09更新
|
731次组卷
|
3卷引用:湖北省部分市州2022-2023学年高二上学期元月联考数学试题
解题方法
10 . 我们知道,在平面中,给定一点和一个方向可以唯一确定一条直线.如点
在直线l上,
为直线l的一个方向向量,则直线l上任意一点
满足:
,化简可得
,即为直线l的方程.类似地,在空间中,给定一点和一个平面的法向量可以唯一确定一个平面.
(1)若在空间直角坐标系中,
,请利用平面
的法向量求出平面
的方程;
(2)试写出平面
(A,B,C不同时为0)的一个法向量(无需证明),并证明点
到平面
的距离为
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1dab74e16403e8131f9f5b2a74f3a84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f41c46212d6f61fca9ce215a477ea1d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdfc3eef2f592a4e93a6968c7f31e32f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e463b86ed390c317de2383840fde5df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a24f3197942ff7bd44f44651dd9123b2.png)
(1)若在空间直角坐标系中,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f3ad64b23e508734de034ce16e1ebbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22ce50ba5e349425274f05d46d120a74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22ce50ba5e349425274f05d46d120a74.png)
(2)试写出平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a46e2fbcd9ba92ca62a67fef9d9652db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab9f353152c7f589c0caf5f964f803ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39f20004bf3d4eb52ec732d8acc65672.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e878d6f51b5830bd59f0d44aa5d8b38.png)
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