1 . 若数列
满足①
,②存在常数
与
无关),使
.则称数列
是“和谐数列”.
(1)设
为等比数列
的前
项和,且
,求证:数列
是“和谐数列”;
(2)设
是各项为正数,公比为q的等比数列,
是
的前
项和,求证:数列
是“和谐数列”的充要条件为
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4ff128b22e3abab2ba8f4d3012c33ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0e19364ae7c7c56b2c27ffb7da111d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e7207cae2e50fa00d0f79ca5d61aaa8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/055ac1dc6b870acc5d654f412e7f8ef2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c88a7ef007c78a93e33bd77c4396626.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c36fc83ff481935ce22168c809930545.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0519af3f19dde038fab2e68b5e2a5387.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0519af3f19dde038fab2e68b5e2a5387.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4c9b799432a92e9fa83ecdba7183f36.png)
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2 . 定义在
上的函数
满足:①当
时,
;②
.设关于
的函数
的零点从小到大依次为
.若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c83e590fa0b13ad3e9c9756642ef53a3.png)
____________ ;若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e91ceea3259f26d816ae075e519804c0.png)
________________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/870ebc2f7aabb028024894568d749934.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec04e67e37fbf0c78aeb7278a9ae1cb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab67eeeec04700ecb63e22874093036b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74a940e5f45501eba9b07b73b2248bf4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd0985146e8adf8ae20d804cc0873c5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39ab8b4ee0658aea3e2deb33a10103eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c83e590fa0b13ad3e9c9756642ef53a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea5a0fd5a050d2d1ea36737c51fed221.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e91ceea3259f26d816ae075e519804c0.png)
您最近一年使用:0次
2016-12-02更新
|
2025次组卷
|
4卷引用:模拟冲刺过关试卷02-【查漏补缺】2022年高考数学三轮冲刺过关(新高考专用)
(已下线)模拟冲刺过关试卷02-【查漏补缺】2022年高考数学三轮冲刺过关(新高考专用)(已下线)2014届北京市海淀区海淀高三上学期期中考试理科数学试卷2015届湖北省黄冈中学高三上学期期末考试理科数学试卷江苏省南京市第二十九中学2021-2022学年高三上学期12月月考数学试题
3 . 已知数列
满足条件:
,且
是公比为
的等比数列,设
.
(1)求出使不等式
成立的
的取值范围;
(2)求
和
,其中
;
(3)设
,求数列
的最大项和最小项的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e34cb00c6d4df3ced2188d432fbed13a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7706e0dba93c9f25c28bc8b01de44b70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c22c96b4d06bc3172cbeb08f4e8c4d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87f4bd63e2b74640143191889a78edcc.png)
(1)求出使不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2be4dd578c5c6772f47db62f64341d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9427914c98562a61b947f58dae6120d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e7dcca2acb8fb6e6a6933a02e0a130b.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f9be52170086e7dd90bf51444d2f6ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc3fba91a1658afc267db5de90dcc613.png)
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4 . 已知数列
的前
项和为
,数列
的前
项和为
,
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c22ee8c2fe975e5156d224a8dc9fda5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d04a8b7a7595251251b8e0b7e665e8c.png)
,
.若对于任意正整数
,都有
成立,则
的最大值为_____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/065054f4e163585d630aa42cb6323a3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a425978da20cebf8c4c63953579e7b35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4a4465705237c08e4e05d849cb28d0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a425978da20cebf8c4c63953579e7b35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cebc697078886727c14d59173ad44f82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c22ee8c2fe975e5156d224a8dc9fda5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d04a8b7a7595251251b8e0b7e665e8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bc21d3aa3d73897faff10fbafcfcca0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3b89c1c4533540a5b0d34ab99789d5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f1f98fb37e8417e282f0ae247a905c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b981b9e68f080db788b70df422ed866.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ab4717e4827480f0f6f4ded85e52eab.png)
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5 . 已知等差数列{
}的前n项和为Sn,公差d>0,且
,
,公比为q(0<q<1)的等比数列{
}中,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0da3dc6065a345f192468df5f441d9eb.png)
(1)求数列{
},{
}的通项公式
,
;
(2)若数列{
}满足
,求数列{
}的前n项和Tn.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82998e334b4dd8a1c285643342598fd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc0b6222a878d450539b2374906a68f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0da3dc6065a345f192468df5f441d9eb.png)
(1)求数列{
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
(2)若数列{
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c59e7c7a84a4bdb959e95536d0404ceb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b812d4eb1cff49beffb2850fd931db2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c59e7c7a84a4bdb959e95536d0404ceb.png)
您最近一年使用:0次
2017-11-22更新
|
649次组卷
|
3卷引用:人教A版(2019) 选修第二册 实战演练 全册综合验收检测
6 . 已知非空集合
满足
.若存在非负整数
,使得当
时,均有
,则称集合
具有性质
.记具有性质
的集合
的个数为
.
(1)求
的值;
(2)求
的表达式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11cc32abee5dba97c6fc3c3a77049b33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38e5789fc7e66811103c74a08313fef0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/582ad43edf388c096e7704d92340bf75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69ea32c2fdee97066d8e3dd2c6580889.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d4fc8faefb26b233d4aa9dbef043aae.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e1a1611f320c0f358df77aaae3f942.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d4fc8faefb26b233d4aa9dbef043aae.png)
您最近一年使用:0次
2016-12-04更新
|
946次组卷
|
6卷引用:2018年高考二轮复习测试专项【苏教版】专题一 集合与简易逻辑
名校
7 . 各项均为正数的数列
的前
项和为
,已知点
在函数
的图象上,且
.
(1)求数列
的通项公式及前
项和
;
(2)已知数列
满足
,设其前
项和为
,若存在正整数
,使不等式
有解,且
恒成立,求
的值.
![](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/b2aeba7035c6ac1c4aeb153f605be79b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ebd1e91be8f6f55a6ff243e328afe8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8f5a88ce01660df217aaf6303ad6dba.png)
(1)求数列
![](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)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b73184fdab4c60b65a15397d1e063384.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/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ca30d7a02a8c91710ffec5405c58f50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168343c417dd908c5e03c4d186b8561b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
您最近一年使用:0次
8 . 数列
的前
项和
,满足
,
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39feec096f8625e6fdad71731fa799c1.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/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16709a4877ae590a92be4ed8e747773d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39feec096f8625e6fdad71731fa799c1.png)
您最近一年使用:0次
名校
解题方法
9 . 已知数列
的前
项和为
,满足
与
的等差中项为
(
).
(1)求数列
的通项公式;
(2)是否存在正整数
,是不等式
(
)恒成立,若存在,求出
的最大值;若不存在,请说明理由.
(3)设
,
,若集合
恰有
个元素,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.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/ec5feaf77d0fecce25efb6c274000d9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f19f39beb2ca207137f0816d87332197.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
(2)是否存在正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a758ea76e914eb4ce69da4916ccd7769.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2158e6be25de78e8d7c15e8843607c8b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d24057a64130b9f47278219c3926888b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
您最近一年使用:0次
2017-12-20更新
|
546次组卷
|
4卷引用:《2018届优等生百日闯关系列》【江苏版】专题二 第六关 以新定义数列为背景的解答题
(已下线)《2018届优等生百日闯关系列》【江苏版】专题二 第六关 以新定义数列为背景的解答题江苏省如东高级中学2017-2018学年高二上学期期中考试数学试题江西省宜丰中学2023-2024学年高二下学期4月期中考试数学试题河南省郑州市宇华实验学校2023-2024学年高二下学期5月月考数学试题
2019高三·全国·专题练习
10 . 某汽车销售公司在军运会期间推广一款新能源汽车,现面向意向客户推出“玩游戏,送大奖”,活动,客户可根据抛掷骰子的结果,操控微型遥控车在方格图上行进,若遥控车最终停在“胜利大本营”,则可获得购车优惠券.已知骰子出现任意点数的概率都是
,方格图上标有第0格,第1格,第2格,……第50格.遥控车开始在第0格,客户每抛掷一次骰子,遥控车向前移动一次,若抛掷出正面向上的点数是1,2,3,4,5点,遥控车向前移动一格(从
到
),若抛掷出正面向上的点数是6点,遥控车向前移动两格(从
到
),直到遥控车移动到第49格(胜利大本营)或第50格(失败大本营)时,游戏结束.设遥控车移动到第
格的概率为
,试证明
是等比数列,并求
,以及根据
的值解释这种游戏方案对意向客户是否具有吸引力.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e6486784415f3537c9a13556c05d893.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b00f4eb7f1bd2ccefbabf0c1dfa8f69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4792fd59c4ca11ff03dc32e367c3983f.png)
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