已知数列
中,
,设数列
满足:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02d36540b44f6c6e8cae00573927f6be.png)
(1)求证:数列
是等差数列,并求数列
的通项公式;
(2)求数列
的通项公式
(3)若数列
满足
,求数列
的前
项和
;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4f4e8b3c380e54988c3212fa02abdde.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02d36540b44f6c6e8cae00573927f6be.png)
(1)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/099a64d86bd0b4602578d910322adc1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(3)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9122df77ac4db7456527103996c3c923.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/08eb71ecf8d733b6932f4680874dbbf3.png)
2021·天津·二模 查看更多[10]
天津市十二区县重点学校2021届高三下学期毕业班联考(二)数学试题(已下线)专题10 数列(难点)-2020-2021学年高二数学下学期期末专项复习(北师大版2019选择性必修第一册、第二册)(已下线)第七章 数列专练10—讨论奇偶(大题)-2022届高三数学一轮复习(已下线)2021年高考数学押题预测卷(天津卷)02(已下线)第4章 数列 单元综合检测(难点)(单元培优)-2021-2022学年高二数学课后培优练(苏教版2019选择性必修第一册)(已下线)第4章《数列》 培优测试卷(三)-2021-2022学年高二数学同步培优训练系列(苏教版2019选择性必修第一册)天津市滨海新区塘沽第一中学2022届高三下学期线上教学调研(一模)数学试题(已下线)数学-2022年高考押题预测卷01(天津卷)天津市第四十七中学2022届高三下学期四月统练数学试题福建省莆田市第五中学2023届高三上学期12月月考数学试题
更新时间:2021-05-01 11:20:02
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解答题-问答题
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名校
【推荐1】已知正项数列
的首项
,前n项和
满足
.
(1)求数列
的通项公式;
(2)若数列
是公比为4的等比数列,且
,
,
也是等比数列,若数列
单调递增,求实数
的取值范围;
(3)若数列
、
都是等比数列,且满足
,试证明: 数列
中只存在三项.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9856bca17b502f3ed74e0640554d6c7a.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00ed46928017a7db641fbb2aad8d0745.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34cead128d88d1c55bab5e5f551d3608.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/161a7b35d1812e6745ae7f7c540cf87a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d7c206dfe5a1c1a300ed801b4a26506.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
(3)若数列
![](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/f1e213371996f2199131d0c3ca8aac88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
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解题方法
【推荐2】已知数列
的前n项和
(
),数列
满足
.
(1)求证:数列
是等差数列,并求数列
的通项公式;
(2)设数列
满足
(
为非零整数,
),问是否存在整数
,使得对任意
,都有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b28ef6f1b2279af482557a8ea46f2e43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49c7cb494cf5ba9bc9c0c961d3b448bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc3ede255fecf6aba650c90309d62670.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/195431ccf2756a0db26f14b7b91a32a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80159bb702d0ec7def697d30b483578f.png)
(1)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/195431ccf2756a0db26f14b7b91a32a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b28ef6f1b2279af482557a8ea46f2e43.png)
(2)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/733969643c55ec0ddfddd781a6545778.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebfc18a8301d182737c78fad0a24ff62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9faeed172ec5b88966b0d1c52748d41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc3ede255fecf6aba650c90309d62670.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9faeed172ec5b88966b0d1c52748d41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc3ede255fecf6aba650c90309d62670.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff84b661009bec09f7a451a3799fef3d.png)
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【推荐1】已知数列
满足
,当
,
时,
.
⑴求数列
的通项公式;
⑵是否存在
,使得
时,不等式
对任意实数
恒成立?若存在,求出
的最小值;若不存在,请说明理由.
⑶在
轴上是否存在定点
,使得三点
、
、
(其中
、
、
是互不相等的正整数且
)到定点
的距离相等?若存在,求出点
及正整数
、
、
;若不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe229b24e2d56ff6b491725ceae4ff2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bc92ab8659c2bad73b8800f626cdc6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1534b035eca39010118a83486de8904.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb8c29b297e3ec337c3139c2a1ebed1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d8a1412b4cb2c9a8b501ec42d95cd6e.png)
⑴求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe229b24e2d56ff6b491725ceae4ff2f.png)
⑵是否存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d6e9f2a9ab9f3612590a52a8e2a2a55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a3b1d1f825ef79e21560dcd6b11c5a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd767e89e58f9ed51a912c704e96e203.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c252a13cd44c516bf1c81cc1e28f957f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
⑶在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ebe7b76deb62e0990b80be0dbca8813.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb80ee96660c424fcc456746f806a95f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/467206b41eccba07bac8269c2dcaec94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e3210a651121500950b779c706de9ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
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【推荐2】已知函数
,各项均不相等的有限项数列
的各项
满足
.令
,
且
,例如:
.
(Ⅰ)若
,数列
的前n项和为Sn,求S19的值;
(Ⅱ)试判断下列给出的三个命题的真假,并说明理由.
①存在数列
使得
;②如果数列
是等差数列,则
;
③如果数列
是等比数列,则
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b84661c468c1df5f538479adc94bb4b.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d902f8881b35b4f310bd5c31ff76e83d.png)
(Ⅰ)若
![](https://img.xkw.com/dksih/QBM/2015/1/7/1571954537906176/1571954543755264/STEM/de7c9f9c14a648f9affb096ca9a60be0.png)
![](https://img.xkw.com/dksih/QBM/2015/1/7/1571954537906176/1571954543755264/STEM/e14b46b175f2488e93d66c5f219b02ac.png)
(Ⅱ)试判断下列给出的三个命题的真假,并说明理由.
①存在数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/983f7e1cd303dc3c2a69ec0aa022f41e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa8fcccbb1234ad67314c96f9856e240.png)
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③如果数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/983f7e1cd303dc3c2a69ec0aa022f41e.png)
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【推荐1】某校开展科普知识团队接力闯关活动,该活动共有两关,每个团队由
位成员组成,成员按预先安排的顺序依次上场,具体规则如下:若某成员第一关闯关成功,则该成员继续闯第二关,否则该成员结束闯关并由下一位成员接力去闯第一关;若某成员第二关闯关成功,则该团队接力闯关活动结束,否则该成员结束闯关并由下一位成员接力去闯第二关;当第二关闯关成功或所有成员全部上场参加了闯关,该团队接力闯关活动结束.
已知A团队每位成员闯过第一关和第二关的概率均为
,且每位成员闯关是否成功互不影响,每关结果也互不影响.
(1)用随机变量X表示A团队第
位成员的闯关数,求X的分布列;
(2)已知A团队第
位成员上场并闯过第二关,求恰好是第3位成员闯过第一关的概率;
(3)记随机变量
表示A团队第
位成员上场并结束闯关活动,证明
单调递增,并求使
的n的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfc321599521a98661ed719cc82ca87c.png)
已知A团队每位成员闯过第一关和第二关的概率均为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cfa1e7ffae662aefb49a44c52d4954d.png)
(1)用随机变量X表示A团队第
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a2c1905706959a99c1962ce75e95d8f.png)
(2)已知A团队第
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec6b15d9089613091e9b5f6108f3d180.png)
(3)记随机变量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29c4cbb3a50014fa18fab2e0de87ee22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e75e8665f4987edbf6a62590564235b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab9310b9dc777d33abe9c873a0c729b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38ab9b42bb564a497bdd5cf14d03d8ca.png)
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【推荐2】在等差数列
中,
,
.各项均为正数的等比数列
的首项为1,其前
项和为
,且
.
(1)求
与
;
(2)设数列
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,
,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37933cfc60b4bd29f1684687ddd2cbd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8845fa12dcd54c2cbe9317726833f605.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/6d9136234a6e2ae1abd445c9907c5c63.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
(2)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07e81f396a80c094690d431557e6cc1e.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72200dad1470cd842739ff1e0e3fb59e.png)
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