名校
解题方法
1 . 已知数列
、
满足
,
.
(1)若数列
为等差数列,求数列
的通项公式;
(2)若数列
是公比2的等比数列,求数列
的前
项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c5a7a17a394e868e0acd1803a9ab795.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/792705b149bd8b752858c060dc67fbc2.png)
(1)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/092aebb9b86d397caa4fd9308b35ad57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
您最近一年使用:0次
2 . 在直角坐标平面内有线段
,已知点
是线段
上靠近
的三等分点,点
是线段
上靠近
的三等分点,……,点
是线段
(
,
)上靠近
的三等分点,设点
的横坐标为
.
(1)求证:数列
为等比数列;
(2)若
,
,求
的通项公式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/473913c0887bb64d386f4c02f1853452.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04b56e44e4f0424a2b7a45567120a2e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/473913c0887bb64d386f4c02f1853452.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e47cd514b2920609e3781c87df6ab70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11603c89c66f064b263af841dae023f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04b56e44e4f0424a2b7a45567120a2e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039af66355ed85ff4c204931b882b694.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f614bf2bc35c4aaf1123d829f7fa82dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3cfeacc29e6a61c5b3b4e439c0a91df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3cfeacc29e6a61c5b3b4e439c0a91df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
(1)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d82c65a855b1eed9c43e6829f6c3bffb.png)
(2)若
![](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/83cf38189d5cbf627d2b82ac0eb76006.png)
您最近一年使用:0次
名校
解题方法
3 . 欧拉函数
表示不大于正整数
且与
互素(互素:公约数只有1)的正整数的个数.已知
,其中
,
,…,
是
的所有不重复的质因数(质因数:因数中的质数).例如
.若数列
是首项为3,公比为2的等比数列,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80c47847486c103fabb5b4ba4220c6a8.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbc89a53c03cb86fb653bb82128f6cba.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/8ac535d98f300ff35496c66fe3c66a5d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8be646cd52d7f2f1714e7542e75810f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adad9633b73dfbbb3d84b4f15979e99e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5b9bd3d8d836eb723be002c86a53740.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b9fa588c3a8ac4df1b963a1f2850163.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80c47847486c103fabb5b4ba4220c6a8.png)
您最近一年使用:0次
2024-06-03更新
|
588次组卷
|
3卷引用:浙江省绍兴市第一中学2024届高三下学期5月模拟数学试题
名校
4 . 已知数列
,其前n项和为
,若存在常数
,对任意的
,恒有
,则称
为
数列.则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e3ad780e73e005ac25acefcab98a37f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2480f87a11c4cd450bc9454ea7276722.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/655c46b33730f3a29b9ec3024df71375.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74ace3f70621a281e73a4ddb168b5e67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e3ad780e73e005ac25acefcab98a37f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a67dad7337d3daaaae14dfb7ea8591fa.png)
A.若![]() ![]() ![]() ![]() |
B.若![]() ![]() ![]() ![]() |
C.若![]() ![]() ![]() ![]() |
D.若![]() ![]() ![]() ![]() |
您最近一年使用:0次
解题方法
5 . 已知
的数列
满足
,
,
成公差为1的等差数列,且满足
,
,
成公比为
的等比数列;
的数列
满足
,
,
成公比为
的等比数列,且满足
,
,
成公差为1的等差数列.
(1)求
,
.
(2)证明:当
时,
.
(3)是否存在实数
,使得对任意
,
?若存在,求出所有的
;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/111d1a60e77d0293acdc3ea1c647d892.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29a7054cf2f1fefdcea1bb11d966cd8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f339d05a6032c0ca8c4187e75d8ae156.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f339d05a6032c0ca8c4187e75d8ae156.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73c0a2ab7198ec8e80904285ca6eb762.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cbbadf02a2855e91a86dedc7a98781a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59dd6c97d2ee3e74ba5730f1cbcc1d43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/306f3c49c9e05cfafadff14fdf90c3f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/965e8beb4ffed1c9cb0110b7e3f580f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8f51bf9165826c40663d01427c24aba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8f51bf9165826c40663d01427c24aba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56c0ec55d00d28d1a877e6ea38d6cd69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e3875830b3121133833a3b45d3407b6.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f65fc200f10b97588a0c9896277c9c64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f6714682274c31a328bf796e235900.png)
(2)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c881b38e5e74dba689507bde6dfa3c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e87d6c4b41cede82adf564ecb513f326.png)
(3)是否存在实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/209559aca6bf32705588b6a40e0b7320.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63b6c614a413bd1db7b6de3a8ff7e7d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
您最近一年使用:0次
名校
解题方法
6 . 欧拉函数
的函数值等于所有不超过正整数
且与
互素的正整数的个数,例如:
,
,
,数列
满足
.
(1)求
,
,
,并求数列
的通项公式;
(2)记
,求数列
的前
和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd22bd64204bf1ca3b9ca6ee0bda60e3.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/9edd29e22f6a7f4d14d9f8d2684d47e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39de1bc04496b97dcf401c669e6ab44e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4979e8653dab16e8eff499e327acffc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8feacd08cf0111e8fa1a62f647bb7f2d.png)
(1)求
![](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/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fef9b91e9ff58a2ccf9ca4f4d88418f9.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)
您最近一年使用:0次
2024-04-19更新
|
2432次组卷
|
6卷引用:浙江省天域全国名校协作体2023-2024学年高三二模数学试题
浙江省天域全国名校协作体2023-2024学年高三二模数学试题山东省青岛第二中学2024届高三下学期二模考试数学试题(已下线)模块4 二模重组卷 第2套 全真模拟卷(已下线)5.2 等差数列和等比数列(高考真题素材之十年高考)(已下线)5.3 数列的求和问题(高考真题素材之十年高考)河南省郑州市宇华实验学校2024届高三下学期5月月考数学试题
7 . 若在数列的每相邻两项之间插入此两项的和,形成新的数列,再把所得数列按照同样的方法不断构造出新的数列.现对数列1,2进行构造,第一次得到数列1,3,2;第二次得到数列1,4,3,5,2;依次构造,第
次得到的数列的所有项之和记为
.
(1)设第
次构造后得的数列为
,则
,请用含
的代数式表达出
,并推导出
与
满足的关系式;
(2)求数列
的通项公式
;
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfc321599521a98661ed719cc82ca87c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
(1)设第
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fc8a581e6abf1cb8f7186e7afb5082e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fcdf86b75a31b39cbc8df2b27164098.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46fc692827ffb41809f7f5417a5a3726.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/090426eb29836bc30c006b3739c08057.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/090426eb29836bc30c006b3739c08057.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
(2)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09c63b276f26170491748a8d8aca0c7c.png)
您最近一年使用:0次
2024-04-13更新
|
422次组卷
|
2卷引用:浙江省舟山市舟山中学2023-2024学年高二下学期4月清明返校测试数学试题
解题方法
8 . 已知
分别是数列
的前
项和,
,
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/478421b81927e435cbcf5acafa89efd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62e567d7e9761951a266953c8d5042ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/758e98bb08ee2d4105904e20c610b421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1376a07ade2c60c5c3bf12886d9487f.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
您最近一年使用:0次
2024-04-03更新
|
328次组卷
|
3卷引用:浙江省杭州市富阳区场口中学2023-2024学年高二下学期3月教学质量检测数学试题
浙江省杭州市富阳区场口中学2023-2024学年高二下学期3月教学质量检测数学试题福建省泉州市2023-2024学年高二上学期1月期末教学质量监测数学试题(已下线)专题01求数列通项公式9种常考题型归类【好题汇编】-备战2023-2024学年高二数学下学期期末真题分类汇编(北师大版2019选择性必修第二册)
名校
解题方法
9 . 数列
满足:
是等比数列,
,且
.
(1)求
;
(2)求集合
中所有元素的和;
(3)对数列
,若存在互不相等的正整数
,使得
也是数列
中的项,则称数列
是“和稳定数列”.试分别判断数列
是否是“和稳定数列”.若是,求出所有
的值;若不是,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62e567d7e9761951a266953c8d5042ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e4b7a84d1c2089430f5adbf0f52731e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2788ae6bae8e954fb96b9a3393adc19.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a55e03428497ac0ea2aa80fe5bdcd939.png)
(2)求集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bb0582b0e415def42abf1d0a567dccb.png)
(3)对数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f04755f109e1dc24e89113809280ece.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/932441c3185a1e55e2dfda8ba7f1e419.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/62e567d7e9761951a266953c8d5042ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7600d2cfbdc6146db96cc545706004f6.png)
您最近一年使用:0次
2024-03-22更新
|
1428次组卷
|
5卷引用:浙江省温州市2024届高三第二次适应性考试数学试题
浙江省温州市2024届高三第二次适应性考试数学试题黑龙江省双鸭山市第一中学2023-2024学年高二下学期4月月考数学试题(已下线)第一套 艺体生新高考全真模拟 (二模重组卷1)湖北省黄冈市文海大联考2024届高三下学期临门一卷(三模)数学试题(已下线)专题07 数列通项公式与数列求和--高二期末考点大串讲(人教B版2019选择性必修第三册)
10 . 用
表示不超过
的最大整数,已知数列
满足:
,
,
.若
,
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3f7fda69e2b32b9ced2239f915fa59b.png)
________ ;若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71c3fdbffff67f73a4f36da898396813.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/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95f9ca737b137a45f33a4cd1d25713c9.png)
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2024-03-14更新
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5卷引用:浙江省强基联盟2024届高三下学期3月联考数学试题