1 .
表示正整数a,b的最大公约数,若
,且
,
,则将k的最大值记为
,例如:
,
.
(1)求
,
,
;
(2)已知
时,
.
(i)求
;
(ii)设
,数列
的前n项和为
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4562f3225c98cf5cb11b47d98c9cc9c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6294a700967de01e6877d686a0e2e79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4160299bf93e7827b97bc5cbb224958e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4c95177c5f6454d2de54bb7b0c182ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae4b8114fcc770a8512cf03da137ca4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9edd29e22f6a7f4d14d9f8d2684d47e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5491950d23d0f3833de05cc3892cacd.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64a7f848e0002222e3fe290e50301e3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ccc57e5668f2a2c1cbc078a767b6855.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16edf0bda2c47ed55f471a1838cd03dc.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/853030075597faf459bec65cd5e0b910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b8178596507fe45cea77096a53d6395.png)
(i)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ce2bf4a86671ab5cefa4d523d8a0fa2.png)
(ii)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/beafadba27d9c078bae7761a2b383803.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47b61920582cc3edd43e273e0cbfa1d4.png)
您最近一年使用:0次
2024-03-26更新
|
1830次组卷
|
8卷引用:重庆市第十一中学校2023-2024学年高三第九次质量检测数学试题
重庆市第十一中学校2023-2024学年高三第九次质量检测数学试题福建省泉州市2024届高三质量监测(三)数学试题广东省佛山市顺德区第一中学西南学校2023-2024学年高二下学期第一次月考数学试卷四川省成都市实验外国语学校2023-2024学年高二下学期第一次阶段考试数学试题辽宁省重点高中沈阳市郊联体2023-2024学年高二下学期4月月考数学试卷(已下线)模块五 专题3 全真能力模拟3(人教B版高二期中研习)(已下线)模块四专题6重组综合练(四川)(8+3+3+5模式)(北师大版高二)(已下线)压轴题05数列压轴题15题型汇总-3
2024高二下·全国·专题练习
解题方法
2 . 高斯是德国著名数学家,近代数学的奠基者之一,享有“数学王子”的称号,用他名字定义的函数称为高斯函数
,其中
表示不超过
的最大整数,如
,
,已知数列
满足
,
,
,若
,
为数列
的前
项和,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc74f388d1672074d66ca67581388f6c.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/2d54a0e82778f606d95a486835ac9f56.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21f2323cbdf0b1b71092c962ae705102.png)
![](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/d1928c254cfada1f75a5cd1e34db5a63.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37d845281cd834068104af1b1aa6027c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c73c39be7e317460e2fe1d4e05195bcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4bc88dbf8fc854838ea57a24924d080.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25c3ac959bdf1b78cb98d92b87c91c46.png)
A.2023 | B.2024 | C.2025 | D.2026 |
您最近一年使用:0次
3 . 等差数列
的前
项和为
,
(
且
),
.
(1)求
的通项公式与前
项和
;
(2)记
,当
,
时,试比较
与
的大小;
(3)若
,正项等比数列
中,首项
,数列
是公比为4的等比数列
,且
,求
的通项公式与
.
![](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/e7fab51121848ce166035ceab6f4e00b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22dd8b3dc4c609bab82d356a5cc2208d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20849c00c47cbdc43f18d53341b6c4e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec3c9f09e932ad3cc1f3720d93b34c56.png)
(1)求
![](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)
(2)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1dcf4e0dd72f254d9eab961b892ae9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf83e20035c3afd6d26ebfd53d768a70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15520cf5be7c2685975aac51bc99ac4f.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e258ab9e600435b37465092243d99f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/385275d29d8c8a7841eaeaa3dfab2cdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7afef6271af7462ffa935a1846e3ec90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89ba85f74cda4ddd621278e558bc036f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3ee3a47bf7dfe538ae367cc006a69b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ea2efac902bfb513a224c40faef2516.png)
您最近一年使用:0次
名校
解题方法
4 . 约数,又称因数.它的定义如下:若整数
除以整数
得到的商正好是整数而没有余数,我们就称
为
的倍数,称
为
的约数.设正整数
共有
个正约数,即为
,
,
,
,
.
(1)当
时,若正整数
的
个正约数构成等比数列,请写出一个
的值;
(2)当
时,若
,
,
,
构成等比数列,求正整数
的所有可能值;
(3)记
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87484a879f450ab097f720fb2a0f4a2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.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/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68c0cd13ec90e5697013e59d73d3e82c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afeed05dbd9752dd537a06bbcbc867cf.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcbd5bb726a08c308b48373afebbb768.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbeaed9ec21e090defafcfeefe0059c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe164d8a8a4049e01565b576007651de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01416ee1d48b17f889e444b7eda99740.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b95a49832d7c33597639bea9eace7989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a57e391b1d575796894fea80cce6329b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04bc6dcaef3c78886e21f1c41e7f2cd6.png)
您最近一年使用:0次
2024-05-04更新
|
168次组卷
|
12卷引用:北京市通州区2023届高三上学期期末数学试题
北京市通州区2023届高三上学期期末数学试题北京市第五十五中学2024届高三上学期10月月考数学试题北京市东城区第六十五中学2024届高三上学期12月月考数学试题广东省广州市广东实验中学2023-2024学年高三下学期教学情况测试(二)数学试卷A湖南省长沙市雅礼中学2024届高三一模数学试卷(已下线)高考数学冲刺押题卷02(2024新题型)(已下线)微考点4-1 新高考新试卷结构压轴题新定义数列试题分类汇编(已下线)专题06 数列(已下线)第四套 艺体生新高考全真模拟 (一模重组卷)湖南省常德市第一中学2023-2024学年高二下学期第一次月考数学试题北京市西城区北京师范大学第二附属中学2023-2024学年高二下学期期中考试数学试题(已下线)高二下学期第三次月考模拟卷(新题型)(范围:导数+选择性必修第三册)-2023-2024学年高二数学题型分类归纳讲与练(人教A版2019选择性必修第三册)
名校
解题方法
5 . 已知数列
的通项公式为
,若对任意
,不等式
恒成立,则实数
的取值范围是______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aed05011f108dad0b561425463d5aae9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e145b6046bc80d0ffecc61ac67c87ca1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03d5596a2e2c7d1cc91c37d397939027.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
您最近一年使用:0次
2024-02-14更新
|
469次组卷
|
4卷引用:安徽省部分学校2024届高三上学期期末质量检测数学试题
名校
解题方法
6 . 已知为数列
的前
项和,且
,若
,
,
是
的前
项和,求
.
您最近一年使用:0次
2024-01-12更新
|
877次组卷
|
2卷引用:黑龙江省海林市朝鲜族中学2023-2024学年高二上学期第三次考试(期末)数学试卷
名校
解题方法
7 . 已知数列
满足:
,设数列
的前
项和为
,若对于任意的
,不等式
恒成立,则实数
的取值范围为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3e5f4d89b65e9fdf1e802757a14a2ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42bb625f34dd81cf351c588a8cba48b1.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/930bc56406e69b785b37a83d48e36724.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/838ec63c0a8824a60e01cf8b0ebdaad3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
您最近一年使用:0次
名校
解题方法
8 . 设数列
的前n项和为
,已知
,
.
(1)证明数列
为等比数列;
(2)设数列
的前n项积为
,若
对任意
恒成立,求整数
的最大值.
![](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/b13a6e1d671215fc96e4bee3541d1096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a00d92524667782ed663ff3d92e720c.png)
(1)证明数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e68223e959d7b55d4b4f30d94059b69c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e145b6046bc80d0ffecc61ac67c87ca1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
您最近一年使用:0次
2023-12-13更新
|
912次组卷
|
4卷引用:江苏省百校大联考2024届高三上学期第二次考试数学试题
解题方法
9 . 已知数列
满足
,设
,其中
表示不超过
的最大整数,
为数列
的前
项和,若
,则正整数
的取值范围为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1762472356358da673d10c101fcf659b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53e2be79297eebc165a4bffcc05e11c6.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/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6dbb46872f37ac2e64e5046d72583f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d010210c784348bd2f41402eac148321.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
您最近一年使用:0次
10 . “杨辉三角”是二项式系数在三角形中的一种几何排列.从第
行开始,第
行从左至右的数字之和记为
,如
,
,
,
的前
项和记为
,依次去掉每一行中所有的
构成的新数列
、
、
、
、
、
、
、
、
、
、
,记为
,
的前
项和记为
,则下列说法正确的有( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38eec2adafdf745ca8ad514469549abb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.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/5e7df0430db8db9fc354ffdd038fb432.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c996a43ff8843aec0be0a9d0ac0e9ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.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/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
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