名校
解题方法
1 . 已知
是由非负整数组成的无穷数列.该数列前
项的最大值记为
,第
项之后各项
的最小值记为
,
.
(1)若
为
,是一个周期为
的数列(即对任意
,
),写出
,
,
,
的值;
(2)设d是非负整数.证明:
(
)的充分必要条件为
是公差为d的等差数列;
(3)证明:若
,
(
),则
的项只能是
或者
,且有无穷多项为
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c36c4148c78af85e5c41562480a84fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64b009b18d73c19e79a6d6d6650e309b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b623b0c3e8607f3442c87c4ac4014c06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/517a9ba04901f83049080e17e971ba7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31a4cfdd9e07678b0f956f89b287b953.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c36c4148c78af85e5c41562480a84fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/373ca638ab28d1698d0ca2a5a5b9824e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8592a1051a0927bd54d00e26d319553f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42eeef885805aa18e46cf9725c0e3248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1a8d81f40b67ff5d714187185b7fdee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22c3825613df085d82ffdb03ede72b10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dba9c413d5c2589337d1c70c2d3e456.png)
(2)设d是非负整数.证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b8c61017b023911c75e4d404b4785cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bce6187f3f11e0ceead8a645f5f9d32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9847d6f5934b3b18db97298dd4f83c97.png)
(3)证明:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44f9dc35e423accb60225ee1d062d33d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f99489791db717b082bd96abb88c55e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/469e197b1ba72e5d014def3a4b1fc946.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1007a9a47e18a607d487a4d4a9559a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
您最近一年使用:0次
名校
解题方法
2 . 令
.
(1)若
,
,试写出
的解析式并求
的最小值;
(2)已知
是严格增函数,
是周期函数,
是严格减函数,
,求证:
是严格增函数的充要条件:对任意的
,
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f389d2b2ea67586fa18c9362e6621b9.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1723a3672a6d29807750ac7803ec379.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13ddac37d332169a34598a63b4634b46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcafc95a0527841c29a58d4f7d85e232.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcafc95a0527841c29a58d4f7d85e232.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a813b5adbf5c7082561237894ba6d599.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef80b434078bf063f1f4c958be301871.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/447d6f62c09c1d05346fd16a24159f6e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b90551ed750a9e91f39d9b5079d9fef.png)
您最近一年使用:0次
名校
解题方法
3 . 定义向量
的“伴随函数”为
;函数
的“伴随向量”为
.
(1)写出向量
的“伴随函数”
,并直接写出
的最大值
;
(2)求函数
的“伴随向量”
的坐标;
(3)已知
,向量
、
的“伴随函数”分别为
、
,设![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ffa8a04c3f3d20b6bd99c157a33b214.png)
,且
的“伴随函数”为
,其最大值为
.求证:向量
的充要条件为
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96e9354f8346e9004bd73ae7cbbb6f52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65c6f29b2b1955715616003d51d8b77f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65c6f29b2b1955715616003d51d8b77f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96e9354f8346e9004bd73ae7cbbb6f52.png)
(1)写出向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2159dcd5f4b6392b9996f09fdbf0b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(2)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b897300cdb9af626f62530c285ca3a2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b6799b234237333b0efa331d98f0374.png)
(3)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae84220aaed30749054f9a2192bec05f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b6799b234237333b0efa331d98f0374.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c968347eabbb636d20b607a3bcfe0ac3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ffa8a04c3f3d20b6bd99c157a33b214.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b320e8296daa8a992a236766aa8f1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cec64476aaca08de0808afda3618109c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0eb7df298a9364b36e079a61caec815c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ed38da21f937df5020532cc9dd35292.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/224cd491a4b179e1e17f5b1afd85040a.png)
您最近一年使用:0次
4 . 已知数列
是无穷数列,满足
.
(1)若
,
,求
,
,
的值;
(2)求证:“数列
中存在
使得
”是“数列
中有无数多项是1”的充要条件;
(3)求证:存在正整数k,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa5bad0e0832bbf42a12f4efc86cfe0e.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039e4fe671d61e59b96ee525c9df43e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2693734765399876e9e93cdb110231c4.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)
(2)求证:“数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0fb67a858cf21e675a4be5ae0bc49c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b64e4722de7deb7c05ca986166e6eb34.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)求证:存在正整数k,使得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01bded46375833efe7c9143aa80f8d64.png)
您最近一年使用:0次
2020-09-13更新
|
1035次组卷
|
3卷引用:2020届北京市中国人民大学附属中学高三上学期期中模拟统练(七)数学试题
2020届北京市中国人民大学附属中学高三上学期期中模拟统练(七)数学试题上海市复旦大学附属中学青浦分校2020届高三下学期开学摸底数学试题(已下线)专题15 数列不等式的证明 微点6 数列不等式的证明综合训练
5 . 已知
是由正整数组成的无穷数列,该数列前
项的最大值记为
,第
项之后各项
,
,
的最小值记为
,
.
(1)若
为
,
,
,
,
,
,
,
,
,是一个周期为
的数列(即对任意
,
),写出
,
,
,
的值.
(2)设
是正整数,证明:
的充分必要条件为
是公比为
的等比数列.
(3)证明:若
,
,则
的项只能是
或者
,且有无穷多项为
.
![](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/d3cfeacc29e6a61c5b3b4e439c0a91df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/090426eb29836bc30c006b3739c08057.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d39222f0687c9124bddb35544bcc7798.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63f5c583c98a1fd516c6ceaa60b55dec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e37ec9529b548c4ba899cf5bf9138eb.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a662a381e0867ce9d871c7a8e71f0d37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6268630d5e5288048d32f4aa5c8bc02d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c171ff5c2728e7cf00a88f88de14f308.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7a6cd655790dcd7f9dd30df72bcbe01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c95c31e29a88bc9a8a3f929f8318afef.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23275b6970ad4a359cd1585b0df6ad70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
(3)证明:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039e4fe671d61e59b96ee525c9df43e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecd79a3b3f98440fd7f473ce4afc13cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
您最近一年使用:0次