1 . 约数,又称因数.它的定义如下:若整数
除以整数
除得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,记为
,
,…,
,
(
).
(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,求证:
;
(3)记
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a980563e0b5b87479dfd8fffd7b4141.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/68c0cd13ec90e5697013e59d73d3e82c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f255d0395fba51ca2d44293cca42e0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0507f52181b9993785471e68f5ecbf7.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/177e4374fb738c4f13dc58e9025c88e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3395c99f805f92a23446c8eb4105b7e.png)
(3)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/535b55b457cd9ebc8cd3f2f029b59bc4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b19449c9426801af1da7045cb785ccd.png)
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3卷引用:广东省阳江市2024届高三下学期5月模拟数学试题
2 . 已知等差数列
前
项和为
,
,
,数列
前
项的积为
.
(1)求数列
,
的通项公式;
(2)设
,求数列
的前
项和
.
![](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/a8a0eecb5b800fce9ae10aed86ffee62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfb14da5d8ba603dbb53af344a9fd84b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63b3c0b86ddb3898525f1ca5f240a84a.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9221c0c92a526f65533cdc5400767af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15520cf5be7c2685975aac51bc99ac4f.png)
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2023高三·全国·专题练习
3 . 设数列
满足
,
.
(1)证明:
.
(2)设数列
的前n项和为
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ea8d0e50065114b05ef2dc1ea1129cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb2dbc4c08aa70076c1c12daeedcb298.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c31f2f5b97bc76078c101082bb76bb6.png)
(2)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41cf1da18d91f7c98086553d157d1a87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ffb575d8e0365de6ccab0d0645fc78a.png)
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4 . 已知数列
满足
.
(1)求数列
的通项公式;
(2)设
,数列
的前
项和为
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0451746d1582cf16235eef4087c5318.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b3e257a216cbe1a806f171d6dd08ca6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.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/8a206f906597710407346e83d3b0dd56.png)
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2023-05-24更新
|
1431次组卷
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4卷引用:广东省阳江市2024届高三上学期第一次阶段调研数学试题
广东省阳江市2024届高三上学期第一次阶段调研数学试题山东省普通高中2023届高三模拟演练数学试题(已下线)专题11 数列前n项和的求法 微点3 裂项相消法求和(一)(已下线)题型16 11类数列通项公式构造解题技巧
名校
解题方法
5 . 已知数列
满足
,
,记数列
的前
项和为
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13a6e1d671215fc96e4bee3541d1096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cceb2c03455b66b7014b3df7282c01db.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)
A.![]() | B.![]() |
C.![]() | D.![]() |
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2022-11-24更新
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2186次组卷
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15卷引用:广东省阳江市高新区2022-2023学年高二上学期期末检测数学试题
广东省阳江市高新区2022-2023学年高二上学期期末检测数学试题广东省佛山市2021-2022学年高二下学期期末数学试题湖北省黄冈市黄梅国际育才高级中学2022-2023学年高三上学期期中数学试题广东省佛山市第一中学2023届高三上学期第三次月考数学试题河南省安阳市第三十九中学2022-2023学年高二上学期第二次加密考试数学试题广东番禺中学2022-2023学年高二上学期期末数学试题福建省福州第一中学2022-2023学年高二上学期第二学段模块考试(期末)数学试题广东省广州市广东番禺中学2022-2023学年高二上学期期末数学试题(已下线)4.1 数列(2)福建省福州第一中学2022-2023学年高二上学期期末数学试题广东省清远市阳山县南阳中学2022-2023学年高二下学期第一次月考数学试题(已下线)第01讲 数列的基本知识与概念(练习)甘肃省庆阳市华池县第一中学2023-2024学年高二上学期期中考试数学试题安徽省合肥市第一中学2023-2024学年高二上学期1月考数学考试试题(已下线) 第4章 数列单元测试基础卷-2023-2024学年高二数学上学期人教A版(2019)选择性必修第二册
名校
6 . 已知数列
满足
,
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe229b24e2d56ff6b491725ceae4ff2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d754bc529cfab94af50384ef686b191d.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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6卷引用:广东省阳江市第三中学2022-2023学年高二下学期期中数学试题
名校
解题方法
7 . 对于项数为
的有穷数列
,设
为
中的最大值,称数列
是
的控制数列.例如数列3,5,4,7的控制数列是3,5,5,7.
(1)若各项均为正整数的数列
的控制数列是2,3,4,6,6,写出所有的
;
(2)设
是
的控制数列,满足
(
为常数,
).证明:
.
(3)考虑正整数
的所有排列,将每种排列都视为一个有穷数列
.是否存在数列
,使它的控制数列为等差数列?若存在,求出满足条件的数列
的个数;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/332a51673c2ae8b003e5f31b71579617.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(1)若各项均为正整数的数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c8a59672f406ba5b297ecabbba62bbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2156cba674f50d2c546502b4a2ed929.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdf0bd1c527c68be985fa95ff6354fe0.png)
(3)考虑正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5aa2de6deb84e4a63b9797d47033a66.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
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4卷引用:广东省阳江市2022-2023学年高二上学期期中数学试题
广东省阳江市2022-2023学年高二上学期期中数学试题上海市复兴高级中学2022届高三下学期3月练习数学试题(已下线)北京市海淀区2023届高三上学期期末练习数学试题变式题16-212024届高三新高考改革数学适应性练习(九省联考题型)
12-13高三上·浙江宁波·开学考试
8 . 数列
中,若
=1,
=2
+3 (n≥1),则该数列的通项
=________
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/090426eb29836bc30c006b3739c08057.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
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|
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17卷引用:2016-2017学年广东省阳春市一中高二文上学期第一次月考数学试卷
2016-2017学年广东省阳春市一中高二文上学期第一次月考数学试卷(已下线)2013届浙江省宁波四中高三上学期期始考试文科数学试卷(已下线)2013-2014学年广西桂林十八中高二上学期段考理科数学试卷(已下线)2013-2014学年广西桂林十八中高二上学期段考文科数学试卷2014-2015学年山东省乐陵市一中高二上学期期中考试理科数学试卷2016-2017学年湖南益阳市箴言中学高二9月月考数学(文)试卷2016-2017年河南漯河高级中学高二理12月月考数学试卷2016-2017年河南漯河高级中学高二文12月月考数学试卷辽宁省辽河油田第二高级中学高二上学期数学必修五 第二章 数列单元测试【全国百强校】甘肃省兰州第一中学2018-2019学年高二9月月考数学试题人教A版 全能练习 第2课时 等比数列的性质江西省宜春市奉新县第一中学2020-2021学年高一下学期第三次月考数学试题海南省琼海市嘉积中学2021-2022学年高二下学期第一次月考数学试题福建省连城县第一中学2023届高三上学期第一次月考数学试题2006年普通高等学校招生考试数学(理)试题(重庆卷)(已下线)专题12 用“不动点法”求数列的通项公式(已下线)艺体生一轮复习 第六章 数列 第25讲 数列的概念【讲】
9 . 已知数列
的前n项和是
,若
,则
的值为________ .
![](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/2f7b62ad23e8ef1b2d4b34671cf2afc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/151532f02f7519599c0436e47174850a.png)
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2021-02-04更新
|
615次组卷
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4卷引用:广东省阳江市高新区2022-2023学年高二上学期期末检测数学试题
10 . 数列
中,
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42e4487468ab2823d6dbf7f0ebd2eb38.png)
___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1d0807149e86eb966a243ceb6c46b95.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42e4487468ab2823d6dbf7f0ebd2eb38.png)
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