名校
解题方法
1 . 数列
满足
则称数列
为下凸数列.
(1)证明:任意一个正项等比数列均为下凸数列;
(2)设
,其中
,
分别是公比为
,
的两个正项等比数列,且
,证明:
是下凸数列且不是等比数列;
(3)若正项下凸数列的前
项和为
,且
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a0bee75d4d83c0b76421fd87113e4dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(1)证明:任意一个正项等比数列均为下凸数列;
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f67fc95a626251da11649acb5e1706f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c340d7d093dd4a275ffea4b87cd26827.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6268630d5e5288048d32f4aa5c8bc02d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c171ff5c2728e7cf00a88f88de14f308.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3755d7aa870e2f199d6c12264fc9be86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
(3)若正项下凸数列的前
![](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/0002f427eded1721f43d60dd0fd3ffe0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd419dc0a6580ab97777b2cb8fd7cded.png)
您最近一年使用:0次
2024-06-12更新
|
1135次组卷
|
5卷引用:2024届湖北省高三普通高中5月联合质量测评数学试卷
2 . 已知数列
的首项为
,且满足
.
(1)求证:数列
为等比数列;
(2)设
,记数列
的前
项和为
,求
,并证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc6545b8eca1c4223ed701a199a85683.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7643e8b7aa32ebf299048417a94432dc.png)
(1)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/213e22890204937a5dded4436369390f.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a00bfec58504040151e3e2101be245a8.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/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6828a1cf75f19bb74a0e0490bd65c168.png)
您最近一年使用:0次
7日内更新
|
372次组卷
|
2卷引用:广东省江门市第一中学2023-2024学年高二下学期第二次段考数学试题
3 . 设自然数
,由
个不同正整数
构成集合
,若集合
的每一个非空子集所含元素的和构成新的集合
,记
为集合
元素的个数
(1)已知集合
,集合
,分别求解
.
(2)对于集合
,若
取得最大值,则称该集合
为“极异集合”
①求
的最大值(无需证明).
②已知集合
是极异集合,记
求证:数列
的前
项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c23747e7321187323c665a641adb49e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84357e7e15c0b4187ea69a4d555ef171.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e29fd31a1808968790032a671f64be90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9a84d5316c5db87fda9b7d615c9dce6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e29fd31a1808968790032a671f64be90.png)
(1)已知集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ccaf946a034215b8c49c12a1aff7790.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d160c80e7542650f9ac8ff3981548ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11391d21b5da91adc137d57a73c19b83.png)
(2)对于集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84357e7e15c0b4187ea69a4d555ef171.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9a84d5316c5db87fda9b7d615c9dce6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
①求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9a84d5316c5db87fda9b7d615c9dce6.png)
②已知集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84357e7e15c0b4187ea69a4d555ef171.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35e9ca7e90d47d7ee295338bbac2d8d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7477e3d7c54f409ee9905e81c9cbe2f.png)
您最近一年使用:0次
名校
解题方法
4 . 若数列
满足
,其中
,则称数列
为M数列.
(1)已知数列
为M数列,当
时.
(ⅰ)求证:数列
是等差数列,并写出数列
的通项公式;
(ⅱ)
,求
.
(2)若
是M数列
,且
,证明:存在正整数n.使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a07614926587f57bc5f341c4f97f4d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec574b71bbd7671223f8c833c8c8b61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(1)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ec1a744042c32d0a851f98fafaa81f3.png)
(ⅰ)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/362832fa3d3c13c1eafd565349d66dce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/115da54f93de5e89d1e7f443fccb61f8.png)
(ⅱ)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0992722f5002aeafa39d25c6b5f4644b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21085fbd6c4b34588f17fc466c845ffe.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a789a9be1723bfbd38ae538a9f39dc1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4ce64685821c3e55c07f151996ca8c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/446e8a7985d4d3dd95c70dc4aad67861.png)
您最近一年使用:0次
2024-03-25更新
|
1245次组卷
|
3卷引用:天津和平区2024届高三一模数学试题
名校
解题方法
5 . 已知数列
的前
项积为
,且
,
.
(1)求证:数列
是等差数列,并且求其通项公式;
(2)证明:
.
![](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/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb5dc2f2e62f4e01cc8cc0aef12f5738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14835bf3f00139ccec0694d0924db795.png)
(1)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de777c4e44546bcfe26ad5b6bb418052.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1172b950b3a1212ba0f75bd18bb70823.png)
您最近一年使用:0次
名校
解题方法
6 . 选用恰当的证明方法,证明下列不等式.
(1)已知
均为正数,且
,求证:
;
(2)已知
,求证:
.
(1)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0fffbec1fe851795dfdd448bf0d165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5558c083d34cbb0a58d3ce1dc6f5778e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/984faa7ae8bb62ce8157b0b60dc84508.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a0c4c098615c6bc7e6dcf72e5b5201a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e10cc5dd849caccce37fe98a26c598.png)
您最近一年使用:0次
名校
7 . 在正项无穷数列
中,若对任意的
,都存在
,使得
,则称
为
阶等比数列.在无穷数列
中,若对任意的
,都存在
,使得
,则称
为
阶等差数列.
(1)若
为1阶等比数列,
,求
的通项公式及前
项和;
(2)若
为
阶等比数列,求证:
为
阶等差数列;
(3)若
既是4阶等比数列,又是5阶等比数列,证明:
是等比数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7905fd422e78a1d22ff6f11950bc5cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d7f4b6e82924087d9fa4523cd509d06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7905fd422e78a1d22ff6f11950bc5cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41a08caf919ff9fa62e20d91af57c401.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0831836c71efc1b1ffdb73073da2a2dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dc71a2fd8c6b263feea5ff5d6a36121.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
您最近一年使用:0次
2024-03-10更新
|
951次组卷
|
4卷引用:河南省TOP二十名校2024届高三下学期质检一数学试题
名校
8 . 已知数列
的前
项和为
,且满足
.
(1)证明:
.
(2)当
时,求证:
;
(3)是否存在常数
,使得
为等比数列?若存在,求出所有满足条件的
的值;若不存在,请说明理由.
![](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/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36ffccbc83f8f43da8b0afd545d0f1b6.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07d212226826bb1d283046f73311a128.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7aeb9a94e392f6759b18abed89aacc5e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/811dfb86daa654123635a048522ab485.png)
(3)是否存在常数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
您最近一年使用:0次
9 . 证明下列不等式
(1)已知
,
,
,且
,求证:
.
(2)已知
,
,
,求证:
.
(1)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67ca5fd57c2c2fcc3c7a574fdd1467d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cec12441802f71e803efaf2c62ee588.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/751e274e9107d780c39ba9c49d6daefb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/135125d796a469155fc4a22dc6be3d10.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/061813f1ec633c5c4c393c4de7938322.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cad3aaeb5b444feb152378278f68863.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5cc7582a7b091b3f0f5a51325e1d0a1.png)
您最近一年使用:0次
23-24高二上·上海·期末
名校
10 . 定义:对于任意大于零的自然数n,满足条件
且
(M是与n无关的常数)的无穷数列
称为M数列.
(1)若等差数列
的前n项和为
,且
,
,判断数列
是否是M数列,并说明理由;
(2)若各项为正数的等比数列
的前n项和为
,且
,证明:数列
是M数列;
(3)设数列
是各项均为正整数的M数列,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68f165a34038d89623948dbe0a669df0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5612ce06759d0f77ca029d10083f7d1e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(1)若等差数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/feb8cf8df82fd05e5549ce9c1a6f3524.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4818548de2563bc81198611cf3468f13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)若各项为正数的等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2c8a7aaf355cf3ea778c73eea8ae635.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de777c4e44546bcfe26ad5b6bb418052.png)
(3)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/292852a3aa9790d661862ff0b67c8971.png)
您最近一年使用:0次
2024-01-14更新
|
1330次组卷
|
8卷引用:期末真题必刷压轴60题(22个考点专练)-【满分全攻略】2023-2024学年高二数学同步讲义全优学案(沪教版2020必修第三册)
(已下线)期末真题必刷压轴60题(22个考点专练)-【满分全攻略】2023-2024学年高二数学同步讲义全优学案(沪教版2020必修第三册)安徽省六安第二中学2023-2024学年高二上学期期末统考数学试卷广东2024届高三数学新改革适应性训练三(九省联考题型)(已下线)第4章 数列(压轴题专练)-2023-2024学年高二数学单元速记·巧练(沪教版2020选择性必修第一册)湖北省荆州市沙市中学2023-2024学年高二下学期3月月考数学试题(已下线)模块五 专题5 全真拔高模拟5(北师大高二期中)(已下线)模块三专题2 数列的综合问题 【高二下人教B版】(已下线)模块三 专题4 数列的综合问题 【高二下北师大版】