1 . 已知函数
是定义在
上的奇函数,且当
时,
,对于数列
,若
,下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a28068770a85b88b42321cd71ecd3c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd5a05989a8c0f72fa134a31e9dbb1cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73d27e9d6651c189516650fb11301b41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0aae580ebcfce670585ba54023be02ed.png)
A.存在![]() ![]() ![]() |
B.![]() ![]() ![]() |
C.![]() ![]() ![]() |
D.若存在等差数列![]() ![]() ![]() ![]() ![]() |
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2 . 若数列
满足:存在等差数列
,使得集合
元素的个数为不大于
,则称数列
具有
性质.
(1)已知数列
满足
,
.求证:数列
是等差数列,且数列
有
性质;
(2)若数列
有
性质,数列
有
性质,证明:数列
有
性质;
(3)记
为数列
的前n项和,若数列
具有
性质,是否存在
,使得数列
具有
性质?说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e976c0663fa749ca749f99842d21ca03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48dfe427f8841f24337b83a767750352.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db5b653a209622a9136a15c3b11b0a4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e976c0663fa749ca749f99842d21ca03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e87e0860e3f142e7ddd7b45c16b211fa.png)
(1)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039e4fe671d61e59b96ee525c9df43e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f24ba3195cbf220d03a1ef5bfe954f9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fde3c47074b6f1b16af81c3684d04419.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e196cf353f8f832f24be4951a9fefab8.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35d16238329f13aeeb2d13aaf025ba07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/662422cae5190af5fa05475a1e16f2d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5344eadd4711db34e3f935aedd5fb270.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b211310c6b436c4b7c4f38ce483d9b13.png)
(3)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/763b41172fa5f9f9ef85ab59df78bc39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de777c4e44546bcfe26ad5b6bb418052.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e87e0860e3f142e7ddd7b45c16b211fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f6b18b109a656b62fb173680ae99ca7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/763b41172fa5f9f9ef85ab59df78bc39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91b7006e157c36d567488d1c30936700.png)
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3 . 已知
为非零常数,
,若对
,则称数列
为
数列.
(1)证明:
数列是递增数列,但不是等比数列;
(2)设
,若
为
数列,证明:
;
(3)若
为
数列,证明:
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9645bd4d2002993b90ec6d48f9c04f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f4fb301b1cc7b26989dc0155265ce8b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5653b60d16ec4e653518f0562680250.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/5882782c03bbca5ac2140b17224139b9.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/15aa2314541715f6be69251653ebf6da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/446e8a7985d4d3dd95c70dc4aad67861.png)
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4 . 若有穷数列
满足:
,则称此数列具有性质
.
(1)若数列
具有性质
,求
的值;
(2)设数列A具有性质
,且
为奇数,当
时,存在正整数
,使得
,求证:数列A为等差数列;
(3)把具有性质
,且满足
(
为常数)的数列A构成的集合记作
.求出所有的
,使得对任意给定的
,当数列
时,数列A中一定有相同的两项,即存在
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef94592b70bea840c747393959c71b39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c735a110f4cf68dea9133c78e205b43f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf404363fe0f057007b8e8d90a775d90.png)
(1)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/030ef2d631bb39945bb752932146364b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf404363fe0f057007b8e8d90a775d90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc9927f218d1b9cd9d7a8b979da6c669.png)
(2)设数列A具有性质
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf9f50605db5d5f8f3a01ee8e474a112.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88e6153f9e3bfe84d3a61f388c7fa2b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd00e20f967cb2bdce939165abd38440.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1c87acdb6ce8286ea7d256b96801507.png)
(3)把具有性质
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf404363fe0f057007b8e8d90a775d90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b835321cd8b7cf192f9e0af0d2f1239b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe83b9b62b3511e37f9726042964db5e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e57696e6509aebe3a8444525b702050e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ce021a66a6856d5078186cffe13f2c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80baa977f2523242a5a3f9a2ac364ad4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b38760e49cb2b3b7bf23410fc189e93.png)
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