1 . 近日北方地区普遍降雪,某幼儿教师手工课上带孩子们做描述雪花形状的图案: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.反复进行这一过程,就得到一条“雪花”状的曲线.设原正三角形(图①)的边长为1,把图①,图②,图③,图④中图形的面积依次记为数列
的前四项,则数列
的通项公式为_____________ ,如果这个作图过程可以一直继续下去,那么“科赫雪花”的面积将趋近于__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/13/da8c63cf-168f-48dd-8a8f-cecaddb69e71.png?resizew=394)
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2024-01-25更新
|
351次组卷
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3卷引用:河北省石家庄市第二十七中学2024届高三上学期金太阳联考数学试题
河北省石家庄市第二十七中学2024届高三上学期金太阳联考数学试题北京市丰台区怡海中学2023-2024学年高二上学期期末模拟练习数学试题(2)(已下线)考点11 由实际问题探究递推关系 2024届高考数学考点总动员
名校
解题方法
2 . 将正三角形(1)的每条边三等分,并以中间的那一条线段为底边向外作正三角形,然后去掉底边,得到图(2);将图(2)的每条边三等分,并以中间的那一条线段为底边向外作正三角形,然后去掉底边,得到图(3);如此类推,将图(
)的每条边三等分,并以中间的那一条线段为底边向外作三角形,然后去掉底边,得到图
.上述作图过程不断的进行下去,得到的曲线就是美丽的雪花曲线.若图(1)中正三角形的边长为1,则图(
)的周长为__________ ,图(
)的面积为___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aceb113626093e0e431f30fa45c2c444.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/2/cc4bbbc0-a228-404d-981c-94e842b746b2.png?resizew=216)
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2021-08-09更新
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1073次组卷
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6卷引用:广东省2022届高三上学期8月阶段性质量检测数学试题
3 . 以下各说法中:
①已知在边长为
的等边三角形
中, 则
;
②若两非零向量
,若
,则
的夹角为锐角;
③已知大小分别为
与
的两个力,要使合力大小恰为
,则它们的夹角的余弦值为
;
④已知数列
的通项
,其前
项和为
,则使
最小的
值为
.
其中正确说法的有________ .(填写所有正确的序号)
①已知在边长为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faed94a64b2dcfc6801b4fca0f16675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05e919aa6826aa94858c261bf05d0e18.png)
②若两非零向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8ccba3b87a8a48ac3dd5f72d00bdb1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90eb96b770a4bed15037b68a77eb9667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8ccba3b87a8a48ac3dd5f72d00bdb1a.png)
③已知大小分别为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf98e72511232b14b841a5d35a607653.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8e74032b421b3c8e6323bcf6af9aee6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3842600c36ce94156ce8ecb6463a288f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d266a04f3dc7483eddbc26c5e487db.png)
④已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67c4593f37ce523cfa8fb5c102240319.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/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d91e07104b699c4012be2d26160976a2.png)
其中正确说法的有
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名校
4 . 数列
满足
,当
时,
,则是否存在不小于2的正整数
,使
成立?若存在,则在横线处直接填写
的值;若不存在,就填写“不存在”_______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35029deaf02b4c1f3c831877f82d035b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5d0a73f50b3e4583f1c1b6d6bf0d18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8357fd2863b8772e2bd211b0a01d951b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2ebaddcbd91bce1d53e0f3b5ead9be7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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2019-09-23更新
|
444次组卷
|
3卷引用:上海市上海中学2018-2019学年高一下学期期末数学试题
名校
5 . 已知下列四个命题:
①等差数列一定是单调数列;
②等差数列的前
项和构成的数列一定不是单调数列;
③已知等比数列
的公比为
,则“
是单调递减数列”的充要条件是“
”;
④记等差数列的前
项和为
,若
,
,则数列
的最大值一定在
处达到.
其中正确的命题有___________ .(填写所有正确的命题的序号)
①等差数列一定是单调数列;
②等差数列的前
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
③已知等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ca664b1e82da6f50064a76fe118aa80.png)
④记等差数列的前
![](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/b7bba46b546008abf9fcbb0db513385f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a417c950a40d58b01af28d2b9a57510e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ef7ca2b3e8061384501f668e59696a1.png)
其中正确的命题有
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名校
6 . 在数列
中,
,且
.记
,
,则下列判断正确的是__________ .(填写所有正确结论的编号)
①数列
为等比例数列;②存在正整数
,使得
能被11整除;
③
;④
能被51整除.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b28ef6f1b2279af482557a8ea46f2e43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d96eda44551ba886992ba70e427a763e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad555526b66f2ab33ed9a56c252f8023.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f032a80a88d742cbaf00df5f7dff9944.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f30bd12847d6f7608f6778b6d14606d3.png)
①数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e160f5b23b74eb9929d6123b4cee2384.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a425978da20cebf8c4c63953579e7b35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c3b5ad9115eb354803901fc86efd7fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb352310b0ca1b455d9ff8d7cc84b9a8.png)
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2018-02-09更新
|
575次组卷
|
5卷引用:河北省邢台市2017-2018学年高一(上)期末测试数学试题
解题方法
7 . 数列{an}的前n项和是
,若数列{an}的各项按如下规则排列:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76907832d191d941c5cbef31600181e1.png)
…,
…
有如下运算和结论:
①
;
②
;
③数列a1,a2+a 3,a 4+a 5+a 6,a 7+a 8+a 9+a 10,…是等比数列;
④数列a 1,a 2+a 3,a 4+a 5+a 6,a 7+a 8+a 9+a 10,…的前n项和
;
在横线上填写出所有你认为是正确的运算结果或结论的序号________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76907832d191d941c5cbef31600181e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a77cb3bee82d44a5d9a96c9a1252f99f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1dfbfb0232750983b537941de085953.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cc7040ad5dd0ebafac0e9f23bd54812.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/277fb4772f833a5320a601add873d192.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87e62963508712b6782f8d5101296187.png)
③数列a1,a2+a 3,a 4+a 5+a 6,a 7+a 8+a 9+a 10,…是等比数列;
④数列a 1,a 2+a 3,a 4+a 5+a 6,a 7+a 8+a 9+a 10,…的前n项和
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a6d72db96302e3dd395a2f58e7f4c22.png)
在横线上填写出所有你认为是正确的运算结果或结论的序号
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8 . 数列
的前n项和是
,若数列
的各项按如下规则排列:
有如下运算和结论:
①
;
②
;
③数列
,
,
,
,…是等比数列;
④数列
,
,
,
,…的前n项和
;
⑤若存在正整数k,使
,则
.
在横线上填写出所有你认为是正确的运算结果或结论的序号________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.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/7946b20669aec5060b4229086a477316.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7a140699aa48bbe6ce03833c4434bac.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87e62963508712b6782f8d5101296187.png)
③数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2466735503515350fb346f469ecc30b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0151a2e204a0664939cb6cb3a5242e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1fcff7f96b0c2c95cd646ad04dc4232.png)
④数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2466735503515350fb346f469ecc30b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0151a2e204a0664939cb6cb3a5242e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1fcff7f96b0c2c95cd646ad04dc4232.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a6d72db96302e3dd395a2f58e7f4c22.png)
⑤若存在正整数k,使
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b762efc9157017b74c2306b2b56c1198.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05cb38daf06c2adeca7841ff3de9d125.png)
在横线上填写出所有你认为是正确的运算结果或结论的序号
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9 . 龙曲线是由一条单位线段开始,按下面的规则画成的图形:将前一代的每一条折线段都作为这一代的等腰直角三角形的斜边,依次画出所有直角三角形的两段,使得所画的相邻两线段永远垂直(即所画的直角三角形在前一代曲线的左右两边交替出现).例如第一代龙曲线(图1)是以
为斜边画出等腰直角三角形的直角边
、
所得的折线图,图2、图3依次为第二代、第三代龙曲线(虚线即为前一代龙曲线).
、
、
为第一代龙曲线的顶点,设第
代龙曲线的顶点数为
,由图可知
,
,
,则
___________ ;数列
的前
项和![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04c2864e2ec3416cc4c081ac1f71a0af.png)
___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/473913c0887bb64d386f4c02f1853452.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4841a7238ffb7413e715d0dfde3c15f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a7469dfbc8ceaec60ecf05a696e5ff7.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/04b56e44e4f0424a2b7a45567120a2e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13a6e1d671215fc96e4bee3541d1096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4266c478e7b7c642a10d37c24896a703.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f92fbacd0a1a4a2f3f5094ece399e34.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42e4487468ab2823d6dbf7f0ebd2eb38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6a8eef8182b33a4f2514f87296d4a9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04c2864e2ec3416cc4c081ac1f71a0af.png)
![](https://img.xkw.com/dksih/QBM/2022/1/21/2899349213429760/2902165343936512/STEM/308c0d3c-6468-458e-bd4e-e1316e61bbf6.png?resizew=682)
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