解题方法
1 . “刺绣”是一门传统手工艺术,我国已有多种刺绣列入世界非遗文化遗产名录.有一种刺绣的图案由一笔画构成,很像汉字“回”,称为“回纹图”(如图). 某刺绣工在方格形布料上用单线针法绣回纹图,共进行了
次操作,每次操作在前一次基础上向外多绣一圈(前三次操作之后的图案分别如下图) . 若第
次操作之后图案所占面积为
(即最外围不封口的矩形面积,如
),则至少操作_______ 次,
不少于
;若每横向或纵向一个单位长度绣一针,称为“走一针”,如图①共走了
针,如图②共走了
针,如图③共走了
针,则其第
次操作之后的回纹图共走了______________ 针(用
表示).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/6/16/7bcd9e4f-65dc-4557-b3fe-5a8868c917f5.png?resizew=516)
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2 . 在算术三角形(也叫帕斯卡尔三角形)中,每个元素(不在第一列)是其正下方的数与左下方的数的差,如图所示,则第四行第五个数是__________ ;若第三行对应的数列记为
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3f7fda69e2b32b9ced2239f915fa59b.png)
__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3f7fda69e2b32b9ced2239f915fa59b.png)
![](https://img.xkw.com/dksih/QBM/2022/6/6/2995545683542016/2995732123000832/STEM/92ebe379-9f11-43e5-8fd2-743cfa21937d.png?resizew=188)
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3 . 如图,在平面直角坐标系中一系列格点
,其中
.且
.记
,如
记为
,
记为
,以此类推.设数列
的前n项和为
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afd21f4cb498101d26b4aaa2e1a6addc.png)
______ ;![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e4b5a382f920203b9ef307224ae641e.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f70db67d96a5bf6d5c6b93ed64952d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73cb3f429dc959e68516e97618d05247.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4519502325b17b37015cb33858826de9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/796d2645ff431d92de68c06bc7fce212.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3032022198160740d6aee8e740d5337.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd5359c8fdc022d7044ffb6fdb291666.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
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![](https://img.xkw.com/dksih/QBM/2022/6/4/2994294233849856/2995642902200320/STEM/9f0f719d-fca5-45da-a1c3-eb88dbb1ab01.png?resizew=267)
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3卷引用:阳光桦树2022年普通高等学校招生统一考试押题卷理科数学试题
4 . 下面几种推理中是演绎推理的为( )
A.某年级有21个班,1班51人,2班53人,三班52人,由此推测各班都超过50人 |
B.猜想数列![]() ![]() ![]() ![]() ![]() |
C.由平面三角形的性质,推测空间四面体性质 |
D.两条直线平行,同旁内角互补,如果![]() ![]() ![]() |
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5 . 如图是一种科赫曲线,其形态似雪花,又称雪花曲线.其做法是:从一个正三角形(记为
)开始,把每条边分成三等份,然后以各边的中间线段为底边,分别向外作正三角形,再把此中间线段去掉,得到图形
;把
的每条边三等份,以各边的中间线段为底边,向外作正三角形后,再把此中间线去掉,得到图形
;依此下去,得到图形序列
,
,
,…,
,….设
的边长为1,图形
的周长为
.给出以下四个结论:①
;②
;③
既有对称轴,也有对称中心;④若
,则n的值最接近于16.以上正确结论的序号是______ .(参考数据:
,
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/635ccd929471d564cc9d2d96266b34d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e9a724b59c890095baa5cb73e267c44.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e9a724b59c890095baa5cb73e267c44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9275bd8ce17fcc4a786510b008414ab0.png)
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![](https://img.xkw.com/dksih/QBM/2022/5/30/2990608080601088/2993214108377088/STEM/983e0b66-c3b7-48ed-9fba-4e01b691e963.png?resizew=205)
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2卷引用:北京市大兴区兴华中学2022届高三三模数学试题
名校
6 . 数学源于生活,数学在生活中无处不在!学习数学就是要学会用数学的眼光看现实世界!1906年瑞典数学家科赫构造了能够描述雪花形状的图案,他的做法如下:从一个边长为2的正三角形开始,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(如图①、②、③等).反复进行这一过程,就得到雪花曲线.
![](https://img.xkw.com/dksih/QBM/2022/5/31/2991153445167104/2992848616611840/STEM/39d311b3-f799-49ca-a6ae-f2e1710e5625.png?resizew=368)
不妨记第
个图中的图形的周长为
,则
( )
![](https://img.xkw.com/dksih/QBM/2022/5/31/2991153445167104/2992848616611840/STEM/39d311b3-f799-49ca-a6ae-f2e1710e5625.png?resizew=368)
不妨记第
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bf64a53be628a9aa2f24113eb17b37c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dbd9ff686703cad03aa383e5fec21.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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9卷引用:北京市第八十中学2022届高三下学期考前热身数学练习试题
北京市第八十中学2022届高三下学期考前热身数学练习试题黑龙江省实验中学2021-2022学年高二下学期期末数学试题(已下线)专题12 数列(已下线)专题17 数列综合应用-3(已下线)模块五 倒数第8天 数列(已下线)第4讲 等比数列的通项及性质5大题型总结(3)(已下线)第五篇 向量与几何 专题20 分形几何 微点2 分形几何综合训练黑龙江省牡丹江市第二高级中学2023-2024学年高三上学期第二次阶段性考试数学试题甘肃省定西市临洮中学2023-2024学年高二上学期第三次质量检测数学试试题
7 . 定义:对于任意一个有穷数列,第一次在其每相邻的两项间都插入这两项的和,得到的新数列称之为一阶和数列,如果在一阶和数列的基础上再在其相邻的两项间插入这两项的和称之为二阶和数列,以此类推可以得到n阶和数列,如
的一阶和数列是
,设它的n阶和数列各项和为
.
(1)试求
的二阶和数列各项和
与三阶和数列各项和
,并猜想
的通项公式(无需证明);
(2)若
,求
的前n项和
,并证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c77a27ecc68192b122861b8c4689ce29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35d0a8da5206f1114ead419f47b81044.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(1)试求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c77a27ecc68192b122861b8c4689ce29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6899bf9cadae2ccdb14cbc87d4f280ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7244499d5babf433375d0b71a672a927.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f640e13b3a3880bf49a49845eee47f07.png)
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名校
8 . 如图(1),画一个边长为1的正三角形,并把每一边三等分,在每个边上以中间一段为一边,向外侧凸出作正三角形,再把原来边上中间一段擦掉,得到第(2)个图形,重复上面的步骤,得到第(3)个图形,这样无限地作下去,得到的图形的轮廓线称为科赫曲线.云层的边缘、山脉的轮廓、海岸线等自然界里的不规则曲线都可用“科赫曲线”的方式来研究,这门学科叫“分形几何学”.
![](https://img.xkw.com/dksih/QBM/2022/5/26/2987761958379520/2991491863502848/STEM/b1ad699b-9df8-4f2e-8dfd-b8511b6c10fc.png?resizew=374)
设第(n)个图形的周长为
,则
与
的递推关系式为______ ,当
时,n的最小值为______ (参考数据:
,
)
![](https://img.xkw.com/dksih/QBM/2022/5/26/2987761958379520/2991491863502848/STEM/b1ad699b-9df8-4f2e-8dfd-b8511b6c10fc.png?resizew=374)
设第(n)个图形的周长为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/090426eb29836bc30c006b3739c08057.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32da1e1a50ec8fab97ff2839736af955.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4998993bfa5c8861a360196735ceb28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fab1cb23df1e01a5120207dbfb4ae6c9.png)
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6卷引用:2022届河南省开封市部分学校高三下学期押题理科数学试题
2022届河南省开封市部分学校高三下学期押题理科数学试题河南省兰考县第一高级中学2022届高三考前押题卷理科数学试题河南省开封市部分学校2022届高考考前押题文科数学试题(已下线)专题20 科赫曲线吉林省吉林市吉化第一高级中学校2022-2023学年高三上学期12月月考数学试题(已下线)第五篇 向量与几何 专题20 分形几何 微点2 分形几何综合训练
9 . 对于数列A:
,经过变换T:交换A中某相邻两段的位置(数列A中的一项或连续的几项称为一段),得到数列
.例如,数列A:
经交换M,N两段位置,变换为数列
:
.设
是有穷数列,令
.
(1)如果数列
为3,2,1,且
为1,2,3.写出数列
;(写出一个即可)
(2)如果数列
为9,8,7,6,5,4,3,2,1,
为5,4,9,8,7,6,3,2,1,
为5,6,3,4,9,8,7,2,1,
为1,2,3,4,5,6,7,8,9.写出数列
;(写出一组即可)
(3)如果数列
为等差数列:2015,2014,…,1,
为等差数列:1,2,…,2015,求n的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adc250de2317c83a904f0ebce5fc2989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24bbf1fc7753df7951558e56b71b78c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adc250de2317c83a904f0ebce5fc2989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a403f562ca2b40d07069aace053e8af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7356ec98b600ece41f3a6b4bc26a7d59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79a60554299801b8c94f80e04398e589.png)
(1)如果数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7356ec98b600ece41f3a6b4bc26a7d59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
(2)如果数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7356ec98b600ece41f3a6b4bc26a7d59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5002f030017f6f0b34a61b2e15c5a9cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bfefd19874aeff8340735b3df468333.png)
(3)如果数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7356ec98b600ece41f3a6b4bc26a7d59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3cfeacc29e6a61c5b3b4e439c0a91df.png)
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10 . 有以下真命题:已知等差数列
,公差为d,设
是数列
中的任意m个项,若
①,则有
②.
(1)当
时,试写出与上述命题中的①,②两式相对应的等式;
(2)若
为等差数列,
,且
,求
的通项公式.
(3)试将上述真命题推广到各项为正实数的等比数列中,写出相应的真命题,并加以证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88890331030408cc74e45ed5083d5478.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c56caecbff605ab9fd337aa8fca99681.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ab1031a0c7db08519d6b56729fc7e2c.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/084d11164c24f8a6d7986123aad2fea2.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ec9d70bd95df12e9e566f1f75092250.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d96983ba858d2d6ebfeb5cdc3b2875c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)试将上述真命题推广到各项为正实数的等比数列中,写出相应的真命题,并加以证明.
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