1 . 将正方形
分割成
个全等的小正方形(图1、图2分别给出了
的情形),在每个正方形的顶点各放置一个数,使位于正方形
的四边及平行于某边的任一直线上的数都分别依次成等差数列.若顶点
处的四个数互不相同且和为1,记所有顶点上的数和为
,则有
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/022205c82846dfa3420c67402a278fea.png)
______ ,…,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d64205bb7d470602fb525b02e72a81b.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/188ba12941ea122a7775a4999e129161.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d15d4afcc5736bb747eeffcffd3f4464.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c82a10b4f0c9323d726804c89dd9548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e345e86daf74312a6992e5d1c3f47f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cf8c7f5f6a35f7ed88271a077619aef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/022205c82846dfa3420c67402a278fea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d64205bb7d470602fb525b02e72a81b.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/5/29/8a0dc8d3-051d-4666-9a45-8e1c699ed7bd.png?resizew=254)
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2 . 下图是瑞典数学家科赫在1904年构造的能够描述雪花形状的图案,图形的作法是:从一正三角形开始,把每条边三等分,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,反复进行这一过程,就得到一条“雪花”状的曲线.若第1个图中的三角形的面积为1,则第
个图形的面积为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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3 . 若
表示自然数
的最大奇因数,例如
,
,
,记
(
为自然数),则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72253d846d8750db2bf695df99c53f3e.png)
______ .,
的通项公式为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e91eb1a74ed4eb789a5cf6bf0d08900a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30ac85888a9b802aea6eb688edac8f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18443df3018a6806c7ded2bf11ed70a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9bf2837ae4419c83bb69ad10ab7ef14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/640faa4d58105334dafd1cf218f30ae9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72253d846d8750db2bf695df99c53f3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38fcec7af3520884b173b29bda6c657a.png)
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4 . 如图,将正整数按下表的规律排列,把行与列交叉处的那个数称为某行某列的元素,记作
,如第2行第4列的数是15,记作
,则有序数对
是____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acb574762e9dc1df3fc79fa957b1fd49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0485cc492f0f7fd84375fd26d4d12585.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49bd66006400c3413106caa2fc94fa2d.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/7/14/8d8501a9-764d-4edf-86bc-031e42d95659.png?resizew=185)
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名校
5 . 阿基米德螺线广泛存在于自然界中,具有重要作用.如图,在平面直角坐标系xOy中,螺线与坐标轴依次交于点
,
,
,
,
,
,
,
,并按这样的规律继续下去.给出下列四个结论:
①对于任意正整数
,
;
②存在正整数
,
为整数﹔
③存在正整数
,三角形
的面积为2023;
④对于任意正整数
,三角形
为锐角三角形.
其中所有正确结论的序号是_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22d07a71ea5e77168d101526bd081433.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca037c7d1fe65ecce1b461a85e81e66.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e47e9f1dbb824847e8a2263c53e908f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb45212d451f476d6841e92293dc02da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfb355a93c360dc25647040e21e5b5e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a1101dbc7953e33c516d7cc282c7842.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b995d143c9835dbc38fb9d7c73e3f8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f8911915f1df38893bddbd642aaee24.png)
①对于任意正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03b15d53749aac7fe0989915ec7db710.png)
②存在正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62f2f63fd60956219cc9eac437aebb3d.png)
③存在正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad64a12f4337c1ce6236b9b2aae324b3.png)
④对于任意正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad64a12f4337c1ce6236b9b2aae324b3.png)
其中所有正确结论的序号是
![](https://img.xkw.com/dksih/QBM/editorImg/2023/8/16/0c32e1c8-c090-4e07-a043-6f16a3f9ef66.png?resizew=170)
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2023-07-10更新
|
401次组卷
|
2卷引用:北京市第二中学2022-2023学年高二下学期第六学段(期末)考试数学试题
6 . 如图,国际象棋棋盘,由64个黑白相间的格子组成,棋盘上2个不同的正方形格如果有一条公共边,就称它们为相邻的.将棋盘上
个白色正方形格作上标记,使得板上的任意黑色正方形格都与至少一个作上标记的白色正方形格相邻,则
的最小值为____________ .若棋盘由
个黑白相间的格子组成,则
的最小值为_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1df05012627be49ae1db0cfd76383f5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/6/27/5baecf2f-025b-4237-8353-d706e3135e13.jpg?resizew=160)
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7 . 如图中的三角形称为谢尔宾斯基三角形,每个图都是取前一个图中的每个黑色三角形三边的中点将其分成四个小三角形,并将中间三角形变为白色,白色三角形不变.若第一个三角形的面积为1,第n个图中白色部分的面积记为
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3f7fda69e2b32b9ced2239f915fa59b.png)
______ .著名的洛卡斯数列
满足
,
,
,
中所有既是偶数,又是3的倍数的项从小到大排列构成一个新的数列,该数列的第n项为
,则数列
的前n项和![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04c2864e2ec3416cc4c081ac1f71a0af.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3f7fda69e2b32b9ced2239f915fa59b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15ab233fde57c65ad8591abac0f6a370.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e1a25c00e9653b88ec05ac86bd86ac7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8903a7918cdf1a954fb699243f150242.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/caa98ea8cb9b47e3c698c606b65689ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15ab233fde57c65ad8591abac0f6a370.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/155e9d6bd5789408a2fd8b04bf08f1d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec4bdc2a6d4fc387dc621f0b5a268c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04c2864e2ec3416cc4c081ac1f71a0af.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/25/8adf23b7-b4bb-46bb-8ea9-2715d47be9da.png?resizew=484)
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8 . 正方形
位于平面直角坐标系上,其中
,
,
,
.考虑对这个正方形执行下面三种变换:(1)
:逆时针旋转
.(2)
:顺时针旋转
.(3)
:关于原点对称.上述三种操作可以把正方形变换为自身,但是
,
,
,
四个点所在的位置会发生变化.例如,对原正方形作变换
之后,顶点
从
移动到
,然后再作一次变换
之后,
移动到
.对原来的正方形按
,
,
,
的顺序作
次变换记为
,其中
,
.如果经过
次变换之后,顶点的位置恢复为原来的样子,那么我们称这样的变换是
-恒等变换.例如,
是一个3-恒等变换.则3-恒等变换共________ 种;对于正整数
,
-恒等变换共________ 种.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a26e951d6d2369e8da79a793a93a66a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e87b3d349194312a934fced615e563c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe280beba9666c98fb0be7c460dd81a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b96608330ee74ee791defc52cf18616d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c88d9142df6ba8e43c1a93bd04a1362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c02b54dc6b3e1bb6544f47d4c8743fcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c02b54dc6b3e1bb6544f47d4c8743fcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.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/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c832f2474efe89961ef41e884da7660c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5030685d4bfdaba51d78d4678f3e101c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc30165c18de623d0a3efb961e606d1c.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/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f255d0395fba51ca2d44293cca42e0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83b16f87e8263bd47e1ca8ccf5206edb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4511c524165a922d034a733ff59ddf6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/940cd957b9db25326a0c7a413df7f1d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f65f59ef448e32a03222f7890e0e611.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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2023-05-19更新
|
876次组卷
|
4卷引用:2023届高三新高考数学原创模拟试题
2023届高三新高考数学原创模拟试题福建省厦门第一中学2022-2023学年高二下学期期末考试数学试题浙江省湖州市第二中学2024届高三下学期新高考模拟数学试题(已下线)第四章 数列(压轴题专练,精选28题)-2023-2024学年高二数学单元速记·巧练(人教A版2019选择性必修第二册)
9 . 对20进行“乘以2”或“减去3”的一种运算,得到的结果再进行“乘以2”或“减去3”的一种运算,…,一直进行这样运算,每进行一种运算记作一次运算,已知运算n次后,得到结果为49,则n的最小值为____________ .
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名校
10 . 阿基米德螺线广泛存在于自然界中,具有重要作用.如图,在平面直角坐标系
中,螺线与坐标轴依次交于点
,
,
,
,
,
,
,
,并按这样的规律继续下去.给出下列四个结论:
,
为整数;②对于任意正整数
,
为整数;③存在正整数
,三角形
的面积为2025;④存在正整数
,三角形
为锐角三角形.其中所有正确结论的序号是________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17584c9b8f4e48c3a8e0a7c8ac626d0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/920e36afc62923f2dda82631490fc547.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3ac934c7b049be27290784a9478c82e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1511466ac46376133963f9276381a83b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/118c102d0a174e5521ec58a2c357bb7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cefa8b3307f16206f2792bfe611e034d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40e1334862f58291d998975c5f571c01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/098eaf2a270d549f7e292d199c8b2906.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62f2f63fd60956219cc9eac437aebb3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a9e6722727ce60728fe1115f645a2fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad64a12f4337c1ce6236b9b2aae324b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db2f40d15b6e2e0d09d4f12d4f600e53.png)
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