1 . 我国元代数学家朱世杰在他的《四元玉鉴》一书中对高阶等差数列求和有精深的研究,即“垛积术”.对于数列
,①,从第二项起,每一项与它前面相邻一项的差构成数列
,②,称该数列②为数列①的一阶差分数列,其中
;对于数列②,从第二项起,每一项与它前面相邻一项的差构成数列
,③,称该数列③为数列①的二阶差分数列,其中
按照上述办法,第
次得到数列
,④,则称数列④为数列①的
阶差分数列,其中
,若数列
的
阶差分数列是非零常数列,则称数列
为
阶等差数列(或高阶等差数列).
(1)若高阶等差数列
为
,求数列
的通项公式;
(2)若
阶等差数列
的通项公式
.
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2506385d68e133523a24a5f5770adb4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f230a901381bb98bd400c14317e0da8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/780007875adc41be137fd9ff68c255b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b21e02ca8a3d50e257ddc00ca87a0406.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e1f74af47ea73ea7fa4e19a51166244.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b397162165301246a7616800610ea6d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/648eaff4ff716932fdbab7ee616b914d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26e024470c4aa889689aefdf14fafddb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
(1)若高阶等差数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a628a7921f38cb09c818b3135aea1a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84f8542101a1cbbf29cfc7a7358a552c.png)
(ⅰ)求的值;
(ⅱ)求数列的前
项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c698639cdba709641d3c91ea1798abfa.png)
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2 . 复平面是人类漫漫数学历史中的一副佳作,他以虚无缥缈的数字展示了人类数学最纯粹的浪漫.欧拉公式可以说是这座数学王座上最璀璨的明珠,相关的内容是,欧拉公式:
,其中
表示虚数单位,
是自然对数的底数.数学家泰勒对此也提出了相关公式:
其中的感叹号!表示阶乘
,试回答下列问题:
(1)试证明欧拉公式.
(2)利用欧拉公式,求出以下方程的所有复数解.
①
;②
;
(3)求出角度
的
倍角公式(用
表示,
).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5aa584db159b0f9bfae801d0134393b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a7035cd4adda5d72a9fc9f9fda75995.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/041a7c8fc017f596542c5e6ec7d1c40b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/574f94ac7dfd3477b58799e0251bb6a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a260aee25664815506d2720174b03829.png)
(1)试证明欧拉公式.
(2)利用欧拉公式,求出以下方程的所有复数解.
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bde2a8df1f0418c41a6e077c7f5de21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1150e58bbcb15a349fb5b9b5ef708d41.png)
(3)求出角度
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f9d7bbcbeb05fbbb06463120f9a6811.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aefd06c239145a2b6ae87a955aa51414.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8cd112c1cb203187e3c9554617c45b8.png)
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名校
解题方法
3 . 对于
,
,
不是10的整数倍,且
,则称
为
级十全十美数.已知数列
满足:
,
,
.
(1)若
为等比数列,求
;
(2)求在
,
,
,…,
中,3级十全十美数的个数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0b1cfbfdf8e1b22aab9583e12e3449c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53f0e26992724eafcba06d163d9ff470.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4217b1854fee34983372bf4f3a877d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5873c01192b7d33b7483f444f90b5b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cdf53108bee755f5aa9a34ea4d163e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5c2b5e218eb815213d8bc0ce9a06ca5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ac416116febcf793fee4ccc78a27b15.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a0f62daf8552adeb241c9b54a57cd83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(2)求在
![](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/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f11075f2c574b6c59b97fb3038000e38.png)
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2024-05-14更新
|
793次组卷
|
6卷引用:山东省泰安市2024届高三下学期高考模拟((三模))数学试题
4 . 莫比乌斯函数在数论中有着广泛的应用.所有大于1的正整数
都可以被唯一表示为有限个质数的乘积形式:
(
为
的质因数个数,
为质数,
),例如:
,对应
.现对任意
,定义莫比乌斯函数
(1)求
;
(2)若正整数
互质,证明:
;
(3)若
且
,记
的所有真因数(除了1和
以外的因数)依次为
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e046acc0e785892df1ef03a440b0fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb5c607987b73502db63f77c9799f4bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08fe943e1acfb453f41bee79119cce60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38261aad19184a74c797b6b88ffd344d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86cb09df4dbbe40a2b7ed54da17346dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09881de0dc186bbcd1e60eb00159ee97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b5872b44498c348c023828ed66e86d1.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9b2c4263428e2ee419589171f27e23f.png)
(2)若正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0fffbec1fe851795dfdd448bf0d165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/201e0fbcfb6833c4b1917cfed3096b6f.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10e468312d09c6563c9094b710a35a65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a4a887eaea7f0aac8505ed3b3c0c678.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/d4811e3603e8790c25aaf91c41d7c7f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/202a57af91d5be04e95fcbdb8f2b788f.png)
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2024-03-26更新
|
1289次组卷
|
5卷引用:湖南省衡阳市2024届高三第二次联考数学试题
湖南省衡阳市2024届高三第二次联考数学试题河南省南阳市西峡县第一高级中学2023-2024学年高二下学期第一次月考数学试卷重庆市乌江新高考协作体2023-2024学年高二下学期第一阶段学业质量联合调研抽测(4月)数学试题(已下线)压轴题08计数原理、二项式定理、概率统计压轴题6题型汇总(已下线)【人教A版(2019)】高二下学期期末模拟测试A卷
名校
5 . 设二项展开式
的整数部分为
,小数部分为
.
(1)计算
,
的值;
(2)求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05889f40a3445ef346269aae7fbbd6f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3cfeacc29e6a61c5b3b4e439c0a91df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63f5c583c98a1fd516c6ceaa60b55dec.png)
(1)计算
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab35850dbc661ded6456b70767cc6cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3087eeade429d61a5daf5b3921f2c95.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f42df3c4d4d6760a1f5705a2d0096e62.png)
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2020-06-30更新
|
621次组卷
|
3卷引用:江苏省扬州中学2020届高三下学期6月模拟考试数学试题
解题方法
6 . 若有穷数列
共有
项
,且
,
,当
时恒成立.设
.
(1)求
,
;
(2)求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc193edab5fe17c89ba2ba6b1b4595f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6281305eb310c4e6138912d15af34e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1339d59bd8ab91843aa455b955af5b8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6593bee36f6d93179bcf9a216cfc1ee5.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9275bd8ce17fcc4a786510b008414ab0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/155c5f3a5c55e0c95191c5a893f63062.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a996c4e29bcc381353e072eb04c11b0.png)
您最近一年使用:0次
7 . 已知函数
.
(1)当
时,若
,求实数
的值;
(2)若
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4f19c89b3d58b2b21534b80179f4df2.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc2d3df37e73a8abea815f37dbb3fff5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c660a2fa2e87189d99aa215342a597c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8250e741d6ee281c4bffaab533ef7de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d24d5828b34d96677a4935168cb1364.png)
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2018-05-08更新
|
1050次组卷
|
2卷引用:【全国市级联考】江苏省苏锡常镇四市2017-2018学年度高三教学情况调研(二)数学试题
解题方法
8 . 在平面直角坐标系xOy中,点P(x0,y0)在曲线y=x2(x>0)上.已知A(0,-1),
,n∈N*.记直线APn的斜率为kn.
(1)若k1=2,求P1的坐标;
(2)若k1为偶数,求证:kn为偶数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f265333edd55c28f04c7dc94b2fa1bba.png)
(1)若k1=2,求P1的坐标;
(2)若k1为偶数,求证:kn为偶数.
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2016-12-04更新
|
953次组卷
|
3卷引用:2016届江苏省南京市高三第三次模拟考试数学试卷
2013·江苏淮安·二模
名校
解题方法
9 . 已知
展开式的各项依次记为
.设函数
.
(1)若
的系数依次成等差数列,求正整数
的值;
(2)求证:
,恒有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7506bbf15ca5a2b36bba7e46f32df84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82fc6513cbb3680c97b6a52dcd17fd51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2951e58ef2f1f504bdb71bdee770bff8.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/909623749d94e2ce3f8873edab20e6d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18fe74b9c8adc168f21a36951d8711d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c9c39fa72f7a96bb1d24a5099ab933f.png)
您最近一年使用:0次
2016-12-04更新
|
570次组卷
|
8卷引用:2013届江苏省淮安市清江附中高三第二次调研测试数学试卷
(已下线)2013届江苏省淮安市清江附中高三第二次调研测试数学试卷江苏省2018年高考冲刺预测卷一数学2016届上海市南洋模范中学高三5月三模数学试题2016届江苏省扬州中学高三上学期12月月考数学试卷专题11.2 二项式定理(练)-江苏版《2020年高考一轮复习讲练测》(已下线)第03讲 二项式定理(核心考点讲与练)-2021-2022学年高二数学下学期考试满分全攻略(人教A版2019选修第二册+第三册)(已下线)专题20 计数原理(模拟练)江苏省徐州市睢宁县第一中学2021-2022学年高二3月学情检测数学试题