名校
解题方法
1 . 南宋的数学家杨辉“善于把已知形状、大小的几何图形的求面积,体积的连续量问题转化为求离散变量的垛积问题”.在他的专著《详解九章算法·商功》中,杨辉将堆垛与相应立体图形作类比,推导出了三角垛、方垛、刍薨垛、刍童垛等的公式. 如图,“三角垛”的最上层有1个球,第二层有3个球,第三层有6个球……第
层球数比第
层球数多
,设各层球数构成一个数列
.
的通项公式;
(2)求
的最小值;
(3)若数列
满足
,对于
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a0876215b2fd463d151523cd3c6b447.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a0876215b2fd463d151523cd3c6b447.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fba64e33de2e9b26c3ecd485a99df0bc.png)
(3)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/538f7dd59772ba33a6fbb271893b1720.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/930bc56406e69b785b37a83d48e36724.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b947eaa62fc4796c9751afbd85f9681.png)
您最近一年使用:0次
2 . 莫比乌斯函数,由德国数学家和天文学家莫比乌斯提出,数学家梅滕斯首先使用
作为莫比乌斯函数的记号,其在数论中有着广泛应用.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)已知
,记
(
为
的质因数个数,
为质数,
,
)的所有因数从小到大依次为
,
,…,
.
(ⅰ)证明:
;
(ⅱ)求
的值(用
(
)表示).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecd9331f692f5f83a74bdba620efe256.png)
![](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/38d94cf780bb9bf7c7da923a99bac6ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e33986442b983a01364b1498d044bbdf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/101edd0628caa05cac88bb6f43788ba6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/367e788c32187ae2cc97aaa24da1d40d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57d45fcbbbc2c58f3aaa95a484df08a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccf9b1f58f95b13bfe77087ed48038a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75cb6c5e6aeca82ba4ab44c352614c35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4ad4926e8bf2b42d8a2c568f80c1987.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/997067e12aa5e1d9b00bb6a9299cb801.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbf3fff8545c74ca66cd1894a55f7bf5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33f39b40e3a5a89d2680d1d47a6bb8e3.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/142df6665826f73a2706e94be482e066.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c16d9bc96f0d4c8992314b315efea8a.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10e468312d09c6563c9094b710a35a65.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/38d94cf780bb9bf7c7da923a99bac6ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e33986442b983a01364b1498d044bbdf.png)
![](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/681ae1522a36768618f7ddaf74abbb7e.png)
(ⅰ)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fa827be71e5fc3cad1b94212d9ed0a6.png)
(ⅱ)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2453fe8eda2466eaf30ce777d60f07b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59c709117ab1d3ef620883a732aed68b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e33986442b983a01364b1498d044bbdf.png)
您最近一年使用:0次
3 . 南宋的数学家杨辉“善于把已知形状、大小的几何图形的求面积,体积的连续量问题转化为求离散变量的垛积问题”.在他的专著《详解九章算法·商功》中,杨辉将堆垛与相应立体图形作类比,推导出了三角垛、方垛、刍薨垛、刍童垛等的公式.如图,“三角垛”的最上层有1个球,第二层有3个球,第三层有6个球……第
层球数是第n层球数与
的和,设各层球数构成一个数列
.
的通项公式;
(2)证明:当
时,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cbad207743c20091cdc5e2114184a01.png)
(3)若数列
满足
,对于
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a0876215b2fd463d151523cd3c6b447.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a0876215b2fd463d151523cd3c6b447.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cbad207743c20091cdc5e2114184a01.png)
(3)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c1ecbdd820cb0c4945e124d29a2b9d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f2360a6dbfca8164cebf81fff5a7282.png)
您最近一年使用:0次
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)
您最近一年使用:0次
2023-03-16更新
|
1019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 . 我国南宋时期的数学家杨辉,在他1261年所著的《详解九章算法》一书中,用如图的三角形解释二项和的乘方规律.此图称为“杨辉三角”,也称为“贾宪三角”.在此图中,从第三行开始,首尾两数为
,其他各数均为它肩上两数之和.
![](https://img.xkw.com/dksih/QBM/2022/1/6/2877405902217216/2945294826274816/STEM/9ae4e6eb2eca4d1494f6798eed7398d1.png?resizew=200)
(1)把“杨辉三角”中第三斜列各数取出按原来的顺序排列得一数列:
,
,
,
,
,…,写出
与
的递推关系,并求出数列
的通项公式;
(2)设
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://img.xkw.com/dksih/QBM/2022/1/6/2877405902217216/2945294826274816/STEM/9ae4e6eb2eca4d1494f6798eed7398d1.png?resizew=200)
(1)把“杨辉三角”中第三斜列各数取出按原来的顺序排列得一数列:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f8c4c029e552954bd493b49aeab82d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d07ae0b4264da6a8812454ffd2f20d94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58b184c94e38f1e5dbe750b2168c2a37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2278c80ff61dc116fa918c177ee4704.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1382989b132b6f41e5a13c2b12805820.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbde65b54e52b3b5d96eeac7f456472c.png)
您最近一年使用:0次
2022-03-27更新
|
502次组卷
|
3卷引用:湖南省长沙市周南中学2021-2022学年高二上学期第三次月考数学试题
解题方法
7 . 设数列
的首项为1,前n项和为
,若对任意的
,均有
(k是常数且
)成立,则称数列
为“
数列”.
(1)若数列
为“
数列”,求数列
的通项公式;
(2)是否存在数列
既是“
数列”,也是“
数列”?若存在,求出符合条件的数列
的通项公式及对应的k的值;若不存在,请说明理由;
(3)若数列
为“
数列”,
,设
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e145b6046bc80d0ffecc61ac67c87ca1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2002a484aeb332ebd6d4d78d4abe5b1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cf5776ec7059c208daf01ca48a34915.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daac43c7675fa411b35028e09b0bad90.png)
(1)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2387880727d458702651d699e76d7d76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)是否存在数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daac43c7675fa411b35028e09b0bad90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/365bd7d058ae516237811aa0d24d9562.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a5d8bc28ee110a9540f383828b7d245.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c9b6e51986fe5d7a7265e0e93adcb4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fc48cdb0d965c38b8765599ee90c299.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3819480e1e624208f729ad8653e4f24f.png)
您最近一年使用:0次
2018-03-06更新
|
1868次组卷
|
6卷引用:江苏省南京师范大学附属中学、天一、海门、淮阴四校2018届高三联考数学调研测试试题
江苏省南京师范大学附属中学、天一、海门、淮阴四校2018届高三联考数学调研测试试题江苏省南师附中等四校2018届高三期初联考数学试题(已下线)《2018届优等生百日闯关系列》【江苏版】专题二 第六关 以新定义数列为背景的解答题(已下线)专题20 与数列有关的恒成立问题-2018年高考数学(理)母题题源系列(江苏专版)【市级联考】江苏省苏北四市2019届高三第一学期期末考试考前模拟数学试题(已下线)专题16 数列新定义题的解法 微点1 数列新定义题的解法(一)
8 . 已知数列1,1,2,1,2,4,1,2,4,8,1,2,4,8,16,
,其中第一项是20,接下来的两项是20,21,再接下来的三项是20,21,22,依此类推. 设该数列的前
项和为
,
规定:若![](https://staticzujuan.xkw.com/quesimg/Upload/formula/188596a6765896c794118d3a39dc0fab.png)
,使得
(![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e84b6d7d85ca0f0bb173f209a909c7c.png)
),则称
为该数列的“佳幂数”.
(1)将该数列的“佳幂数”从小到大排列,直接写出前3个“佳幂数”;
(2)试判断50是否为“佳幂数”,并说明理由;
(3)(i)求满足
>70的最小的“佳幂数”
;
(ii)证明:该数列的“佳幂数”有无数个.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
规定:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/188596a6765896c794118d3a39dc0fab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/858911660b233271d57b17e358232d45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d76a1197aaabd0077aafc8d6e850747d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e84b6d7d85ca0f0bb173f209a909c7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cad135b14c9dcd83eab6618d7694c7b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(1)将该数列的“佳幂数”从小到大排列,直接写出前3个“佳幂数”;
(2)试判断50是否为“佳幂数”,并说明理由;
(3)(i)求满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(ii)证明:该数列的“佳幂数”有无数个.
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2018-01-26更新
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662次组卷
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3卷引用:北京市昌平区2018届高三上学期期末考试数学(理)试题
北京市昌平区2018届高三上学期期末考试数学(理)试题(已下线)微考点8-1 新高考新题型19题新定义题型精选江西省抚州市崇仁一中、广昌一中、南丰一中、金溪一中四校2023-2024学年高二下学期第二次月考数学试卷
9 . 若数列
的前
项和
满足:
,记
.
(1)求数列
的通项公式;
(2)若
,
,求证:
;
(3)记
,求
的值.(注:[x]表示不超过x的最大整数,例:[2.1]=2,[-1.3]=-2)
![](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/c8370a173854471a3eb27637993a3d5d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0f6000421c5370e4b89f23be199f388.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50e1ee88beaddafb0d0a185c3a8e0dc5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5619944e5472bdd87d6889e768eedbdc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/015221d24ded0923094d54cf77450bac.png)
(3)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17fc2e0e9c1aca4b68f4c2ac242b4150.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/183d41343b820104ccb9bb527b9e2b26.png)
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