1 . 2022年第二十四届北京冬奥会开幕式上由96片小雪花组成的大雪花惊艳了全世界,数学中也有一朵美丽的雪花——“科赫雪花”.它的绘制规则是:任意画一个正三角形
,并把每一条边三等分,以三等分后的每边的中间一段为边向外作正三角形,并把这“中间一段”擦掉,形成雪花曲线
.重复上述两步,画出更小的三角形,一直重复,直到无穷,形成雪花曲线
,
,
,
,
.
设雪花曲线
周长为
,面积为
.若
的边长为3,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f512b7e52f01aa7d4afb59287de2a5c8.png)
________ ;![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04c2864e2ec3416cc4c081ac1f71a0af.png)
________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2708fa6298e52f617383efc175b71ddc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b9cb8e6ff801523b0304576cd69fd2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/797e67927616b141ed7c6b83f8b6f4fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fee50575e3ebd56c4f46dd0bbf8e55d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf83e20035c3afd6d26ebfd53d768a70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/30/b3ed9892-cd6b-4c2a-a65d-d6f5c5f8fbc1.png?resizew=305)
设雪花曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf83e20035c3afd6d26ebfd53d768a70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c7c5e7bd6bac51402ffa04b4144ec78.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2708fa6298e52f617383efc175b71ddc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f512b7e52f01aa7d4afb59287de2a5c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04c2864e2ec3416cc4c081ac1f71a0af.png)
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2 . 小王准备在单位附近的某小区买房,若小王看中的高层住宅总共有n层(
,
),设第1层的“环境满意度”为1,且第k层(
,
)比第
层的“环境满意度”多出
;又已知小王有“恐高症”,设第1层的“高层恐惧度”为1,且第k层(
,
)比第
层的“高层恐惧度”高出
倍.在上述条件下,若第k层“环境满意度”与“高层恐惧度”分别为
,
,记小王对第k层“购买满意度”为
,且
,则小王最想买第______ 层住宅.
(参考公式及数据:
,
,
,
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/821759d44fa4f3d3ef5929c9e0f8be91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15a70b95c53fb6655721e2a8c61f5c2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4135ae310b5e3475bb86b4d78fbfc4a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/699dfd96d64e59252e384847629c7a75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/432851e0d0b7a2924da29b9cc5ca1706.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ce99aa5814346374feae45d0c7a5c91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4135ae310b5e3475bb86b4d78fbfc4a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/699dfd96d64e59252e384847629c7a75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/432851e0d0b7a2924da29b9cc5ca1706.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f255d0395fba51ca2d44293cca42e0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/233427826eb2233641fc3a9805f6d206.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9e5dfcc28321b563a8012ec2899c502.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf9ae71d3572a47498db249c6c0759e3.png)
(参考公式及数据:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b7c4ccd5ec8eb377a9d7b664d63030b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0591d9f78b4f4f78c5bd6baaa602ae0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/289ad328bffb5f497153dc0e59939257.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e027147b6f150c3b9e0243defd8b22e.png)
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2023-08-20更新
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5卷引用:湖北省高中名校联盟2024届高三上学期第一次联合测评数学试题
湖北省高中名校联盟2024届高三上学期第一次联合测评数学试题江西省南昌市第十九中学2024届高三上学期第二次月考数学试题江苏省南通市如皋市2024届高三下学期2月诊断测试数学试题(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)(已下线)专题8 函数新定义问题(过关集训)(压轴题大全)
名校
解题方法
3 . 已知无穷等差数列
中的各项均大于0,且
,则
的最小值为___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8121c3de0fc0ed687ca9aef2ab50821.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43be0c3c40c33b7c424e89a10e0a0d41.png)
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2卷引用:湖北省黄冈市浠水县第一中学2024届高三上学期9月质量检测数学试题
4 . 已知三棱锥
的顶点处有一质点M,点M每次会随机地沿一条棱向相邻的某个顶点移动,且向每一个顶点移动的概率都相同,从一个顶点沿一条棱移动到另一个顶点称为移动一次.若质点M的初始位置在点A处,则点M移动2次后仍然在底面ABC上的概率为__________ ,点M移动n次后仍然在底面ABC上的概率为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
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5 . 盒子里装有5个小球,其中2个红球,3个黑球,从盒子中随机取出1个小球,若取出的是红球,则直接丢弃,若取出的是黑球,则放入盒中,则:
(1)取了3次后,取出红球的个数的数学期望为___________ ;
(2)取了
次后,所有红球刚好全部取出的概率为___________ .
(1)取了3次后,取出红球的个数的数学期望为
(2)取了
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/084fb0695a8dd2a2169ab044787fbd67.png)
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4卷引用:湖北省2023届高三下学期5月国都省考模拟测试数学试题
湖北省2023届高三下学期5月国都省考模拟测试数学试题吉林省长春市第二中学2022-2023学年高三下学期第七次调研测试数学试卷(已下线)第七章 随机变量及其分布(压轴题专练)-2023-2024学年高二数学单元速记·巧练(人教A版2019选择性必修第三册)(已下线)专题20 概率与统计常考小题归类(15大核心考点)(讲义)
6 . 黎曼猜想由数学家波恩哈德∙黎曼于1859年提出,是至今仍未解决的世界难题.黎曼猜想研究的是无穷级数
,我们经常从无穷级数的部分和
入手.请你回答以下问题:
(1)![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f1e7df075a8aea5c473b84fbe93b4a6.png)
_____ ;(其中
表示不超过
的最大整数,如
)
(2)已知正项数列
的前
项和为
,且满足
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a96c7a4f80a6323ab9957d1fabe391fc.png)
_________ .(参考数据:
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc07ef1256a9188949462dff0bc9be7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86b3bd282c6e7cad9cf53cde43b122da.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f1e7df075a8aea5c473b84fbe93b4a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40ca5bb1a4a8e02c13874056ccdeb27e.png)
(2)已知正项数列
![](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/9583a4d9bf7b954042226232d23a8c19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a96c7a4f80a6323ab9957d1fabe391fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/227cad9f324ed0089526402e3977f329.png)
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7 . 高斯函数
是以德国数学家卡尔-高斯命名的初等函数,其中
表示不超过
的最大整数,如
.已知
满足
,设
的前
项和为
的前
项和为
.则(1)![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb1f05f5309cef367574296ca026946f.png)
__________ ;(2)满足
的最小正整数
为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7179c645736d68c90023f83d7f11ed01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a34d1fadd6b439e84582c2165dd4124.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e43322b79e09a1863b2e80b1211c756.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b508eca8ccf27db702d436a9c8b2f2e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/576b8a96318251cedee755512e73e7c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc05e75f68b63e0bdf88e06c77bcc07f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb1f05f5309cef367574296ca026946f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a3f82e93615cd6510caa63c5899629c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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8 . 对于数列
,定义
为数列
的“加权和”,已知某数列
的“加权和”
,记数列
的前n项和为
,若
对任意的
恒成立,则实数p的取值范围为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bd85a0ac7f7884ab723cfbe7b600684.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/2bf2a0b2082c017897fa25db1476f198.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e94592f4c368e38ec045e6fee449218f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edbb0cfb0e5d5b41fdf2bf6787e425c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/930bc56406e69b785b37a83d48e36724.png)
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4卷引用:湖北省红安县第一中学2022-2023学年高二下学期3月月考数学试题
9 . 已知
的前
项和为
,
,
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19b7daadaea74c1a9d8f97fd0b4086f1.png)
______ .
![](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/dbc4bef6bd30798531aa087574b96578.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04451d025577f4f82bed48adf90d7299.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19b7daadaea74c1a9d8f97fd0b4086f1.png)
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10 . 在正项数列
中,
,
,记
.整数m满足
,则数列
的前m项和为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e7953a2a3dc1833844e72a9d93ac2f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c08780dd277c645d9bb0587a3303011.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf5eadfe9ceec0c4434371014c6d29c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
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2023-02-09更新
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2卷引用:湖北省部分名校2023届高考适应性考试数学试题