对于项数为
的数列
,若满足:
,且对任意
,
与
中至少有一个是
中的项,则称
具有性质
.
(1)如果数列
,
,
,
具有性质
,求证:
,
;
(2)如果数列
具有性质
,且项数为大于等于5的奇数,试判断
是否为等比数列?并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d95e47ebf576a5b2be0a9e09cca9d864.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18db8b768e5060b3471415e4b55ac30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aeee42a40351adbd5d9445133f0fddfb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abf9fc9e8c9940547678ff7934363f52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(1)如果数列
![](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/daf464629fa321a6ff7401ab79f07083.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54dd31377b22061e084513fb6c532969.png)
(2)如果数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
更新时间:2022-12-28 18:17:34
|
相似题推荐
解答题-问答题
|
较难
(0.4)
名校
解题方法
【推荐1】已知正项数列
的前n项和为
,且满足
,数列
满足
,
,且
.
(1)求数列
与
的通项公式;
(2)求数列
的前
项的
;
(3)将数列
与
的项相间排列构成新数列
,设新数列
的前
项和为
,若对任意正整数n都有
,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b28ef6f1b2279af482557a8ea46f2e43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1cd8ccd6f9a0291fbaead54838fe56e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/195431ccf2756a0db26f14b7b91a32a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f06d1f542ed677ac03bdafe9fa0373b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89dd6a7c5bfa27105d9ee45087a62ab3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f32c3daa8ee6f36dd82d616ad717d95.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b28ef6f1b2279af482557a8ea46f2e43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/195431ccf2756a0db26f14b7b91a32a7.png)
(2)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec4bdc2a6d4fc387dc621f0b5a268c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a425978da20cebf8c4c63953579e7b35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
(3)将数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b28ef6f1b2279af482557a8ea46f2e43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/195431ccf2756a0db26f14b7b91a32a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbf096159d50e95915127352ff57dbb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/733969643c55ec0ddfddd781a6545778.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a425978da20cebf8c4c63953579e7b35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffbaab8d220965c7e7224a5fa1593315.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e2c42014a37da5916a0cba01474984a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13c62e7c58606b2250c6d189d85a6ea8.png)
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【推荐2】设
是各项为正数且公差为d
的等差数列
(1)证明:
依次成等比数列;
(2)是否存在
,使得
依次成等比数列,并说明理由;
(3)是否存在
及正整数
,使得
依次成等比数列,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dec99c57bf7997bd93e1ed8f48d5af9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b60fc6993ed58c125ae54ec70c48c718.png)
(1)证明:
![](https://img.xkw.com/dksih/QBM/2015/6/24/1572144820469760/1572144825868288/STEM/940552c963cf47d6bb21078c31971c6b.png)
(2)是否存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3da2dd8d2a73a50b19abd87bb39f48f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34d1770855f8ff72ed7a4ae34684617f.png)
(3)是否存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3da2dd8d2a73a50b19abd87bb39f48f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac1c4b8f96da2495ecc059119eb01e0f.png)
![](https://img.xkw.com/dksih/QBM/2015/6/24/1572144820469760/1572144825868288/STEM/6a0b3dfefeca4ac5b557f6e1bab1b0c3.png)
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【推荐3】将保护区分为面积大小相近的多个区域,用简单随机抽样的方法抽取其中15个区域进行编号,统计抽取到的每个区域的某种水源指标
和区域内该植物分布的数量
,得到数组
.已知
,
,
.
(1)求样本
的样本相关系数;
(2)假设该植物的寿命为随机变量
(
可取任意正整数),研究人员统计大量数据后发现,对于任意的
,寿命为
的样本在寿命超过
的样本里的数量占比与寿命为1的样本在全体样本中的数量占比相同,均为0.1,这种现象被称为“几何分布的无记忆性”.
(i)求
的表达式;
(ii)推导该植物寿命期望
的值(用
表示,
取遍
),并求当
足够大时,
的值.
附:样本相关系数
;当
足够大时,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97ea8f47d8d8d9e1832d52b1c7425450.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02471a4dd55b13c35d8ffaf7c3717c80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef046c85a536174bec951a53d9f60b33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27b562e9bd801d9b060054dbad4cf8da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2d9cd101a6f493e68226c889cb9eef1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ac6ba0fbab855b55efd132706206c34.png)
(1)求样本
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/725985b2e0488ae470a1d4c86a746dee.png)
(2)假设该植物的寿命为随机变量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7399fcd570d1de4057f2059759d18cc9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b00f4eb7f1bd2ccefbabf0c1dfa8f69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(i)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8647973329cfd3cdf53cc16f24ccac9.png)
(ii)推导该植物寿命期望
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bf3baba074e8aeb6f3ea117865bbd1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4203d1a6c2c250a210b7d5acf02cb4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bf3baba074e8aeb6f3ea117865bbd1b.png)
附:样本相关系数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fff8afd3871b55a19add8c331d18058.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d48723131b516e95795e360967b0a176.png)
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解题方法
【推荐1】若各项为正的无穷数列
满足:对于
,
,其中
为非零常数,则称数列
为
数列.记
.
(1)判断无穷数列
和
是否是
数列,并说明理由;
(2)若
是
数列,证明:数列
中存在小于1的项;
(3)若
是
数列,证明:存在正整数
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f8365233f341451598eb50525a1557a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/782fdf6345302a3d8814acf96f6b3acb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5653b60d16ec4e653518f0562680250.png)
(1)判断无穷数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93068e5f0dedec981ec828ffa4458c4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e4b5779873cb3f4366dbfdb983dec81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/446e8a7985d4d3dd95c70dc4aad67861.png)
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【推荐2】如果一个数列从第2项起,每一项都与它的前一项的差都大于2,则称这个数列为“
数列”.
(1)若数列
为“
数列”,且
,
,
,求实数
取值范围;
(2)是否存在首项为1的等差数列
为“
数列”,且其前
项的和
满足
?若存在,请求出
的通项公式;若不存在,请说明理由;
(3)已知等比数列
的每一项均为正整数,且
为“
数列”,
,
,当数列
不是“
数列”时,试判断数列
是否为“
数列”,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edf1b5b5f5f9269db4e3f748bd7f5348.png)
(1)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edf1b5b5f5f9269db4e3f748bd7f5348.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f73a97d20f5bc6ce114cd7ae7845c009.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7b1bfb98c2ca5473a25db8e422aa3c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92279feebef06abd0302005390123d91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(2)是否存在首项为1的等差数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edf1b5b5f5f9269db4e3f748bd7f5348.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/771d258a9c027fca9daec87acf8c36cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)已知等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edf1b5b5f5f9269db4e3f748bd7f5348.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b16a90364bdffdb10175942d399cc895.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c281045a3511eefe1121aea3a57ce5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edf1b5b5f5f9269db4e3f748bd7f5348.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edf1b5b5f5f9269db4e3f748bd7f5348.png)
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