名校
1 . 阅读材料一:“装错信封问题”是由数学家约翰·伯努利(Johann Bernoulli,1667~1748)的儿子丹尼尔·伯努利提出来的,大意如下:一个人写了
封不同的信及相应的
个不同的信封,他把这
封信都装错了信封,问都装错信封的这一情况有多少种?后来瑞士数学家欧拉(Leonhard Euler,1707~1783)给出了解答:记都装错
封信的情况为
种,可以用全排列
减去有装正确的情况种数,结合容斥原理可得公式:
,其中
.
阅读材料二:英国数学家泰勒发现的泰勒公式有如下特殊形式:当
在
处
阶可导,则有:
,注
表示
的
阶导数,该公式也称麦克劳林公式.阅读以上材料后请完成以下问题:
(1)求出
的值;
(2)估算
的大小(保留小数点后2位),并给出用
和
表示
的估计公式;
(3)求证:
,其中
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66d4e8502106802f1485c3b0f28f2664.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a8412f5256b2b370e421c07f18cc732.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4403d632f9a81e52c6cd135c6834bc2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
阅读材料二:英国数学家泰勒发现的泰勒公式有如下特殊形式:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ce152ca98ac7e21237e00667f005b62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35993bd1db970330494665d925c0be7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(1)求出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/395c6efaa63dcd4ee513323d51c6a7eb.png)
(2)估算
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2598975ac1edb754817eada15b9a473e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/041a7c8fc017f596542c5e6ec7d1c40b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66d4e8502106802f1485c3b0f28f2664.png)
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca08ded0d1136421f0a81517f5c2fc9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
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2 . 在数学中,把只能被自己和1整除的大于1自然数叫做素数(质数).历史上研究素数在自然数中分布规律的公式有“费马数”
;还有“欧拉质数多项式”:
.但经后人研究,这两个公式也有局限性.现有一项利用素数的数据加密技术—DZB数据加密协议:将一个既约分数的分子分母分别乘以同一个素数,比如分数
的分子分母分别乘以同一个素数19,就会得到加密数据
.这个过程叫加密,逆过程叫解密.
(1)数列
中
经DZB数据加密协议加密后依次变为
.求经解密还原的数据
的数值;
(2)依据
的数值写出数列
的通项公式(不用严格证明但要检验符合).并求数列
前
项的和
;
(3)为研究“欧拉质数多项式”的性质,构造函数
是方程
的两个根
是
的导数.设
.证明:对任意的正整数
,都有
.(本小题数列
不同于第(1)(2)小题)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66fa614dd0a4ef38831d742ed3e2c883.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29e8e0703bc265e4b6659d5076564fcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf31876698721a199c7c53c6b320aa86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b09c918f20cda7e931d16ba79baf0020.png)
(1)数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c14d9ae06f864498048d55088ff4e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72c12274ae6ca7bc2d0ad2ced6a0337d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c14d9ae06f864498048d55088ff4e6.png)
(2)依据
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c14d9ae06f864498048d55088ff4e6.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/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(3)为研究“欧拉质数多项式”的性质,构造函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d26cf2a1b49eb3f90d64d7fc526bf4c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86b92b70365c63607daecdc8deb73ecf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a6ce810257873cb94a56a93b39537d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b00e0b2cfc9260694affc6b33f59eb89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6148cff72e9eabbf9912e158b52f0129.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
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2024-05-28更新
|
536次组卷
|
2卷引用:安徽省皖北五校联盟2024届高三第二次联考数学试卷
3 . 在
个数码
构成的一个排列
中,若一个较大的数码排在一个较小的数码的前面,则称它们构成逆序(例如
,则
与
构成逆序),这个排列的所有逆序的总个数称为这个排列的逆序数,记为
,例如,
.
(1)计算
;
(2)设数列
满足
,
,求
的通项公式;
(3)设排列
满足
,
,
,
,
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34e04f64c273928cb099d08ac52cfcf4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc77dfe095330d5ac22696e02745f4f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b066322d5ce7859e174207d32fdeb8e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fb8280885d0fd1a072039e0bbcd15a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50bae0107d95c2964c862d83a78a7880.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c74b667cbad8dc6743f8f267be05880.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb8b82f01d3e473e2eb9cb2d6c74cb74.png)
(1)计算
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe67d956e76fbdc799d356b6fb492c80.png)
(2)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d94669ca9b5a7ad3de1034b7503ca0d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a1404c7e8a894900a5265a502adf478.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)设排列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9c1ded5ba5f43cdcf3e79c56db2f630.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4be0310608bc9ed911cad3df317bddbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37be536781a2cad0ab0721237513cd54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2699a580bcb4b0517f7c055cad6568a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31a5e3db38502800e4c7f999185bba33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f633a299fcefe6528943858cc8a5536c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8154ded0f61fb250cbccccfe9f646ef1.png)
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4 . 某少数民族的刺绣有着悠久的历史,下图①、②、③、④为她们刺绣最简单的四个图案,这些图案都由小正方形构成,小正方形数越多刺绣越漂亮,现按同样的规律刺绣(小正方形的摆放规律相同),设第
个图形包含
个小正方形.
(1)求出
;
(2)归纳出
与
的关系式,并根据你得到的关系式求
的表达式;
(3)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d4fc8faefb26b233d4aa9dbef043aae.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/24/00374c3b-fe2d-4001-b2fb-8c0307bb517e.png?resizew=407)
(1)求出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afc5119887c500b60dd3f3975068a363.png)
(2)归纳出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09d81d4730668cce0bbeb1f8fd39e0b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d4fc8faefb26b233d4aa9dbef043aae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d4fc8faefb26b233d4aa9dbef043aae.png)
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19f16ded042e7b4ab649f860d5b05669.png)
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5 . 高斯是德国著名的数学家,近代数学奠基者之一,享有“数学王子”的称号,以他的名字定义的函数称为高斯函数
,其中
表示不超过x的最大整数.已知数列
满足
,
,
,若
,
为数列
的前n项和.
(1)证明:数列
是等比数列,并求数列
的通项公式;
(2)求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8aacac2cf1dd70cc65b1ca535a32c316.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2ab85825d4a002600ca41bd3cd2ee7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039e4fe671d61e59b96ee525c9df43e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7def23f30138e0b7c4c1e498d6903a6c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e46bca035f977f168c82ad4fce6845bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce49ab12f75d0829be561a7b3ed42a4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3462d4e0565158697bc5a14107f7407c.png)
(1)证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d82c65a855b1eed9c43e6829f6c3bffb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c444c4d2fd7d65c5c1434e44814895b9.png)
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2023-03-16更新
|
1020次组卷
|
4卷引用:上海市宝山区2023届高三下学期3月月考数学试题
上海市宝山区2023届高三下学期3月月考数学试题(已下线)重难点02数列求和的五种解题方法-【满分全攻略】2022-2023学年高二数学下学期核心考点+重难点讲练与测试(沪教版2020选修一+选修二)(已下线)第六篇 数论 专题2 数论函数 微点3 数论函数综合训练上海市延安中学2024届高三下学期3月月考数学试题
6 . 2018年9月24日, 英国数学家M.F阿蒂亚爵在“海德堡论坛”展示了他“证明”黎曼猜想的过程,引起数学界震动. 黎曼猜想来源于一些特殊数列求和, 记![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ce3ffd6593c57695a2d0e3390879f4b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ce3ffd6593c57695a2d0e3390879f4b.png)
A.![]() |
B.![]() |
C.![]() |
D.![]() |
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7 . 2018年9月24日,英国数学家
阿帝亚爵在“海德堡论坛”展示了他“证明”黎曼猜想的过程,引起数学界震动,黎曼猜想来源于一些特殊数列求和,记
,则
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5365bd5f3254f05c369ee09efcccc91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4793ae9a3ca0874fc88457c880fc9571.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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