1 . 已知各项均为整数的数列![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ad107fc0e55e4b35b2b25b10f75f4e6.png)
.满足
,且对任意
,都有
.记
.
(1)若
,写出一个符合要求的
;
(2)证明:数列
中存在
使得
;
(3)若
是
的整数倍,证明:数列
中存在
使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ad107fc0e55e4b35b2b25b10f75f4e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a0ed3f2a79403d4ca1cf2f9def5ae31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e557ad17fa38ac6b1f55e6ad6ec3c31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c906536bb830afee02111d791983e06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b389b37b65b78e0242245f67b5f2dc82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6735b270c7b4dbf195e1834d745e3dd.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13a6e1d671215fc96e4bee3541d1096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f762938f5c78eb72bafbb13bf85cba1.png)
(2)证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0818b44478f6d1d972aa5bf6dd4d3a0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f255d0395fba51ca2d44293cca42e0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/433eaf536c1fed0f48f4af7b595a2af4.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d76da8e15e302756b4d2e7e24906ccb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0818b44478f6d1d972aa5bf6dd4d3a0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5566b1d828acdac47fe50216d247cfac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60cc91f55eeaa0330a9586ee73912466.png)
您最近一年使用:0次
2021-05-07更新
|
1196次组卷
|
9卷引用:北京市朝阳区2021届高三下学期二模数学试题
北京市朝阳区2021届高三下学期二模数学试题北京市朝阳区人大附中朝阳分校2022届高三12月月考数学试题北京市第八中学2021-2022学年高二6月月考数学试题北京市第二中学2023届高三上学期10月月考数学试题北京市第五十七中学2023届高三上学期12月月考数学试题北京市北京理工大学附属中学2023届高三下学期开学测试数学试题北京市第五中学2022-2023学年高二下学期期末检测数学试题【北京专用】专题03数列(第三部分)-高二上学期名校期末好题汇编(已下线)专题01 数列(6大考点经典基础练+优选提升练)-【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(新高考专用)
2 . 设数集S满足:①任意
,有
﹔②对任意x,
(x,y可以取相同值),有
或
,则称数集S具有性质P.
(1)判断数集
和
是否具有性质P,并说明理由;
(2)若数集
且
具有性质P.
(i)当
时,判断
是否一定构成等差数列,说明理由;
(ⅱ)若
,数集B中的每个元素均为自然数且
,求数集B中所有元素的和的所有可能值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c8647b00cc8c8f35555c7d78cf2812c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e2e79843faf62dde86bf858d1e0569.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45c8b2714e2f6ddfdd6b05d3b4de1149.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b8e5872f45d4b878b0119997cb5bae2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84734fbba70c0b45045fabf8090f810b.png)
(1)判断数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e300e8b92758e3f411495dde53d5a904.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/802bb7e2a3ad9686147591de5e12392b.png)
(2)若数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ddc189d2c675b0e2ade4f7ed40f66fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93ef80455ed7f56718cc4f1d8b837b30.png)
(i)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fac3649308b528fd56545ba102dc42d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dec99c57bf7997bd93e1ed8f48d5af9a.png)
(ⅱ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2ab800bb4666f21dbe05ec239ca39ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06f73eec2bbbfa166f874c39d05accb6.png)
您最近一年使用:0次
解题方法
3 . 设有限数列
:
,定义集合
为数列A的伴随集合.
(1)已知有限数列
:-1,0,1,2和数列
:1,2,4,8.分别写出
和
的伴随集合;
(2)已知有限等比数列
:
,求
的伴随集合M中各元素之和S;
(3)已知有限等差数列
:
,判断0,
,
是否能同时属于
的伴随集合M,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e710265624cd25e9ff1cba21049c6b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f64c014274afc8444b5366ed496a09a2.png)
(1)已知有限数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
(2)已知有限等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/235d9f675b6449eda598c2e49f0fd5a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(3)已知有限等差数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb3e9ef023f9cb43f9c9930f3b9620a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3909298815b2f723f4618721d8c0bf1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cb02a09032d9fa0d9353acb21edb1c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
您最近一年使用:0次
4 . 已知无穷数列{an},对于m∈N*,若{an}同时满足以下三个条件,则称数列{an}具有性质P(m).
条件①:an>0(n=1,2,…);
条件②:存在常数T>0,使得an≤T(n=1,2,…);
条件③:an+an+1=man+2(n=1,2,…).
(1)若an=5+4![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2468403b3eba9e40bfa36f464e927738.png)
(n=1,2,…),且数列{an}具有性质P(m),直接写出m的值和一个T的值;
(2)是否存在具有性质P(1)的数列{an}?若存在,求数列{an}的通项公式;若不存在,说明理由;
(3)设数列{an}具有性质P(m),且各项均为正整数,求数列{an}的通项公式.
条件①:an>0(n=1,2,…);
条件②:存在常数T>0,使得an≤T(n=1,2,…);
条件③:an+an+1=man+2(n=1,2,…).
(1)若an=5+4
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2468403b3eba9e40bfa36f464e927738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27cba79639deae5f8af6088b30c1a800.png)
(2)是否存在具有性质P(1)的数列{an}?若存在,求数列{an}的通项公式;若不存在,说明理由;
(3)设数列{an}具有性质P(m),且各项均为正整数,求数列{an}的通项公式.
您最近一年使用:0次
2021-05-02更新
|
1166次组卷
|
5卷引用:北京市海淀区2021届高三下学期期中数学试题
5 . 在数学中,双曲函数是与三角函数类似的函数,最基本的双曲函数是双曲正弦函数与双曲余弦函数,其中双曲正弦:
,双曲余弦函数:
,(e是自然对数的底数)
(1)解方程:
;
(2)写出双曲正弦与两角和的正弦公式类似的展开式:
_________,并证明;
(3)无穷数列
,是否存在实数a,使得
?若存在,求出a的值,若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/444eaaaf560b7ed9b225e9ba34c5dbbd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f980c011d926a51ef8201dc79e88002.png)
(1)解方程:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01958f2959a7fa4a86d8b25058ccb1ba.png)
(2)写出双曲正弦与两角和的正弦公式类似的展开式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e13b62168f8e90860410fde112257aec.png)
(3)无穷数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d04f8e5504b64a6eada18e2dc997b55f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd167bb422b4b23cfc8fe76c9a6c5a41.png)
您最近一年使用:0次
6 . 已知数列
满足
,
,并且
,
(
为非零参数,
).
(1)若
成等比数列,求参数
的值;
(2)当
时,证明:
;
(3)当
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ad5eee51276c36c3b0ec5473504958a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afb6c7c4c02de4c67f60d31ed1139bcc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3706c8575b004154908c34c973feba03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4159df4d2540cc3909c26128314e82e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c77422a29ac2408a030888c50042c64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6ec0b97655e6bd7004df04457c493ac.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bc321d11e01d8b1ef4879278eb385f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3be362dec96173f246ff747264007817.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/323c36bb62b2165b80aa4d388901a086.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76d8047f0a8bd0cf4e250cd0fe80093b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/396c3d8c0d5b9d36dcfcde77865960d8.png)
您最近一年使用:0次
7 . 已知满足递推条件:
,且
,
,求
的通项公式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ebd8dd04868877c21ad59e6fd5966fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/489340c9a2d70c00bae13b7018cad448.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2815b24f5a89be7ae53aed93182e8988.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
您最近一年使用:0次
8 . 定义:对于任意一个有穷数列,在其每相邻的两项间都插入这两项的和,得到的新数列称为一阶和数列,如果在一阶和数列的基础上再在其相邻的两项间插入这两项的和,得到二阶和数列,以此类推可以得到
阶和数列,如
的一阶和数列是
,设n阶和数列各项和为
.
(1)试求数列
的二阶和数列各项和
与三阶和数列各项和
,并猜想
的通项公式(无需证明);
(2)设
,
的前
项和
,若
,求
的最小值
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3448994b1b6e02d0c7fff2a1df678ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d4898ec28df07f52c9caecd36f86307.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(1)试求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3448994b1b6e02d0c7fff2a1df678ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6899bf9cadae2ccdb14cbc87d4f280ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/846fa57d92d6ad44d6a0cafad1e71ed4.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc727f022d27f8b55930798fd387d45a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7e74be91bfe4bc209da7539dbf9b72c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2167179a1ccbdb8093df0580709e629.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
您最近一年使用:0次
名校
解题方法
9 . 设函数
.
(1)求
的值和
的解析式;
(2)是否存在非负实数
,使得
恒成立,若存在,求出
的值,若不存在,请说明理由;
(3)定义
,且
(
),
①当
时,求
的解析式;
②已知下列正确的命题:当
(
,
)时,都有
恒成立;对于给定的正整数
,若方程
恰有
个不同的实数根,确定
的取值范围,若将这些根从小到大排列组成数列
(
),求数列
所有
项的和.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a31712c94832db2eb6ede22d263d7bae.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e335d1d1f5754d72aece814a55cc2841.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4909270f94e2c30e489b2d51499012a.png)
(2)是否存在非负实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85fb3099a99b9397809ac06981589fc7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)定义
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b597680aefd3635872a7adaebb7d3b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ec64aa8b89793ae9e0b84c1b3974d8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15a70b95c53fb6655721e2a8c61f5c2c.png)
①当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00c80887660b9043931cfac788514b7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46261412955df2580730200e19f5ff91.png)
②已知下列正确的命题:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b83a52167b4e0ba9c1a96dfe635c6783.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bca32309e7c22b53659f849edbcb3fe5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b7cd4b73d476e71d831fa9f86477641.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca5e27e05ca489ccd7dbf3e81ae3325.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf87092f371c316b415779cf5a33fed2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e77d6f15137ae5d98b0d546672b6f68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/086e9b14c35ef3c57b20f5e952ebf9c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/510d9423fd34558d0ffcb75e98524de8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/086e9b14c35ef3c57b20f5e952ebf9c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e77d6f15137ae5d98b0d546672b6f68.png)
您最近一年使用:0次
名校
解题方法
10 . 已知函数
,数列
各项均为正数,且数列
、
满足:
,
,
.
(1)设
,
,若
是无穷等比数列,求数列
的通项公式;
(2)若对于给定的
满足
,问:是否存在递减数列
,使得
是无穷等比数列?若存在,求出
的取值范围,若不存在,说明理由;
(3)当
时,
为公差不为0的等差数列且其前
的和为0;若对任意满足条件
的数列
,其前
项的和
均不超过
,求正整数
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b923078510697d5f7f9ea392eb76dd9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/292b7800ba829f3f458cd6c23edf68a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9882a4d12c45d37b650d7e6120c7df7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e31d5f5293de76ffd02c8125caa9eb6.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61f75723824158fae6941098197f51b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9adba66018e60187eeb78d67e13db91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)若对于给定的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b75c671c153751a24549bee59555c7f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64e021a223c85a5fe67380cf5d489ccd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2360f4c62f7f1173922e755529a00fae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00750edf6ba5e93875f77676e7107a9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2360f4c62f7f1173922e755529a00fae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98de51986abddede978630e0db085330.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f08a01aed02ce1eaf1aaefaa0342b7ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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