已知数集
(
,
)具有性质
:对任意的
、
(
),
与
两数中至少有一个属于
.
(1)分别判断数集
与
是否具有性质
,并说明理由;
(2)证明:
,且
;
(3)证明:当
时,
、
、
、
、
成等比数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a98a3d1f11a31e9ab1a3dde94c2d58d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fec089f9d4b5d8a70cb2ecfd8f1678c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7600d2cfbdc6146db96cc545706004f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/937c09d82c480e4d67f8a48d3f66c5f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f52783e7a39f438adf08ef7d05d8c78.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abf9fc9e8c9940547678ff7934363f52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(1)分别判断数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c0c0c967a628666433195b3c356b345.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37a3069c8accda13019e775a5dc198c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4049e63727f73d80c8b0b12b795285ac.png)
(3)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e45cf86650443d1b86c79b1e3edc7e5c.png)
![](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/f65fc200f10b97588a0c9896277c9c64.png)
更新时间:2020-01-01 15:54:57
|
相似题推荐
解答题-证明题
|
较难
(0.4)
名校
【推荐1】设
为正整数,区间
(其中
,
)同时满足下列两个条件:
①对任意
,存在
使得
;
②对任意
,存在
,使得
(其中
).
(Ⅰ)判断
能否等于
或
;(结论不需要证明).
(Ⅱ)求
的最小值;
(Ⅲ)研究
是否存在最大值,若存在,求出
的最大值;若不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c724c6119e3e17b6181178ce7e6baf75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29d1fd5262cae918d9c8ef6a1bede788.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33f84aa794bc075d6139177cd2f59925.png)
①对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bbba3561714a2b7b7b675e4c319e4cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b375f090c551bb2817fa942edbf9bd05.png)
②对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/165df5a77d87e7c534898e995f162562.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bbba3561714a2b7b7b675e4c319e4cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b5de90d938c439d3a9a8e5e1880604f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/927a02889cbfc416da88181520058c3a.png)
(Ⅰ)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/deb6b5ca66b71ac5daa42ce59f19f72d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/432851e0d0b7a2924da29b9cc5ca1706.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f6b3e4ab38102e50c861c13496bd215.png)
(Ⅱ)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
(Ⅲ)研究
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
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名校
【推荐2】已知非空有限实数集
的所有非空子集依次记为
……,集合
中所有元素的平均值记为
.将所有
组成数组
……,数组
中所有数的平均值记为
.
(1)
,求
;(2)
……![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf0b3797c7366b3e8e529b91d597a719.png)
,若
……
,求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcf4e20ea341827ce5f9552daee39462.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d0a9523f2084cf17b8656c11ab1d95e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/233427826eb2233641fc3a9805f6d206.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/233427826eb2233641fc3a9805f6d206.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2e6a2b019a4803f58f9d1357d193b13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91cfa8b06cc2ed7e3323fb726edb815d.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/264f13296143d4cc38ac1e154cba0971.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91cfa8b06cc2ed7e3323fb726edb815d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e23d3511f35774402cdad5e133e6d11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf0b3797c7366b3e8e529b91d597a719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c4a19bc9518862e3bc7c3495657bec5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa152620e0b322267cc6e714ff75b816.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7662283ac4777446531fac7ee97da9a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91cfa8b06cc2ed7e3323fb726edb815d.png)
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【推荐1】已知数列
中,
,
为实常数),前
项和
恒为正值,且当
时,
.
⑴求证:数列
是等比数列;
⑵设
与
的等差中项为
,比较
与
的大小;
⑶设
是给定的正整数,
.现按如下方法构造项数为
有穷数列
:
当
时,
;
当
时,
.
求数列
的前
项和
.
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/ccbaf04054424308a846648b619644ee.png)
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/9860ad5ee2b5416e872477cb3da32b0d.png)
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/6e1e40316de542ddbf4e144fd341d813.png)
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/cf3f41b050cf4cc1aa005d17444941d1.png)
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/51b9f8c888a04a088a27cda1c5e3be0f.png)
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/bfe6c2fd42b24965ba51ad4104833f85.png)
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/c437b9ef7bd741efb33563e39c991cc9.png)
⑴求证:数列
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/9083cce8e9cf44c78b023042b67c4dae.png)
⑵设
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/2dbc456962124d9d8576e12f3d644d24.png)
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/521809270c924c55af7db34fe3dd24b2.png)
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/f8232306d7ce4c328738c85f7c0b22d5.png)
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/f8232306d7ce4c328738c85f7c0b22d5.png)
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/b6f107f6983748dc85bae3610e4ae9c8.png)
⑶设
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/e6deab0d4280479d91d3e677d657662a.png)
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/ed7807dc7ea842b5aee85a1f8a8c99b6.png)
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/f04b5409ec9249e88b19756c7af7d337.png)
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/f484e81d29274951973de2552d90abdf.png)
当
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/d311eaa06a3f4739a3146302a3de120b.png)
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/d6dca304039c45b786d44dae1c3c51c8.png)
当
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/e43a5a82100e49279c5d04430efb693e.png)
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/a34955c063484647bddf7e9ad74ff649.png)
求数列
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/6dc3bed9239147968b2589bbf1cb3465.png)
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/cf3f41b050cf4cc1aa005d17444941d1.png)
![](https://img.xkw.com/dksih/QBM/2012/2/21/1570756940939264/1570756946378752/STEM/f858c7d1160f4a47b33fac74e78720a6.png)
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解答题-问答题
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【推荐2】已知数列{an}的各项均为整数,其前n项和为Sn.规定:若数列{an}满足前r项依次成公差为1的等差数列,从第r﹣1项起往后依次成公比为2的等比数列,则称数列{an}为“r关联数列”.
(1)若数列{an}为“6关联数列”,求数列{an}的通项公式;
(2)在(1)的条件下,求出Sn,并证明:对任意n∈N*,anSn≥a6S6;
(3)已知数列{an}为“r关联数列”,且a1=﹣10,是否存在正整数k,m(m>k),使得a1+a2+…+ak﹣1+ak=a1+a2+…+am﹣1+am?若存在,求出所有的k,m值;若不存在,请说明理由.
(1)若数列{an}为“6关联数列”,求数列{an}的通项公式;
(2)在(1)的条件下,求出Sn,并证明:对任意n∈N*,anSn≥a6S6;
(3)已知数列{an}为“r关联数列”,且a1=﹣10,是否存在正整数k,m(m>k),使得a1+a2+…+ak﹣1+ak=a1+a2+…+am﹣1+am?若存在,求出所有的k,m值;若不存在,请说明理由.
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解答题-证明题
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较难
(0.4)
解题方法
【推荐1】数列
的前
项和为
,且满足
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4140364a99be65199470a74e2b3a1878.png)
(1)设
,求证:数列
是等比数列;
(2)设
,求
的最小值.
![](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/6963786821fc6fbe0f4ec96f351dcd84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4140364a99be65199470a74e2b3a1878.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5fa25a0d5c7358fd1523c741efac978.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98b57460aa38110c05a31e20b9e38c67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c59e7c7a84a4bdb959e95536d0404ceb.png)
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解题方法
【推荐2】已知数列
的前
项和
满足:
.
(1)求证:数列
是等比数列,并求数列
的通项公式;
(2)若数列
满足
,
为数列
的前
项和,求证:
.
![](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/67bd36be3bb1aa5eb5db74b2a7af7f7e.png)
(1)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c485d7f863edc6299df64bd89d4705b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fce83115a50f99e08e9a2db7267aeed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01e2b779d4e1468d0cc9bb859653f618.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de633c277a234e59e274ffb1f9d59718.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cf49305249eb983fb10c95c2287d1ee.png)
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