解题方法
1 . 若实数列
满足
,有
,称数列
为“
数列”.
(1)判断
是否为“
数列”,并说明理由;
(2)若数列
为“
数列”,证明:对于任意正整数
,且
,都有![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce97bb4c969108ebef4ebadd5acc5ca4.png)
(3)已知数列
为“
数列”,且
.令
,其中
表示
中的较大者.证明:
,都有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08fee3639b8e3ff9717f3ffd9633927f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8414472e2121e1796eb40408d820053a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
(1)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81f1bc95528c774ce919f7c5a0ef0d5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4d7e68d0c8bd9a32d826c721ab74d9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22e517c32ed93de4d9cb6e0926337000.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce97bb4c969108ebef4ebadd5acc5ca4.png)
(3)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93846b72089bd425864969f2edabdb8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8957564d30af14db69fdc36be2eaee7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/065f8a641aab5c39f73be89a6f2e7a65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4dc7e8bd20e9d38b4d248a8c253ccd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86e8131eb05d480f0da6c61ebda3ce48.png)
您最近一年使用:0次
名校
解题方法
2 . 集合
,将集合A中的元素按由小到大的顺序排列成数列
,即
,
,数列
的前n项和为
.
(1)求
,
,
;
(2)判断672,2024是否是
中的项;
(3)求
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbe134d52a2548dcb569dd0b133881bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06dd3c3b45125d4b484e2894992610f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69c7266d90661cf4467f13c6f5eb670c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(1)求
![](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)判断672,2024是否是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
(3)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8062be9f491b4357fd3e336f4ae31c54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fb035fe568e25a250beeba0d75b5c59.png)
您最近一年使用:0次
3 . 广渝高速公路指挥部接到预报,24小时后将有一场超历史记录的大暴雨,为了确保万无一失,指挥部决定在24小时内筑一道临时堤坎以防山洪淹没正在紧张施工的隧道工程.经测算,其工程量除现有施工人员连续奋战外,还需要20辆翻斗车同时作业24小时.但是,除了有一辆车可立即投入施工外,其余车辆需要从各处紧急抽调,每隔20分钟能有一辆车到达并投入施工.而指挥部最多可组织25辆车,问24小时内能否完成防洪堤坎工程?并说明理由.
您最近一年使用:0次
解题方法
4 . 已知数列中,
,且
,
为其前
项的和.
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
(2)求满足不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/690ea16d79cbc17cf606fb21ca85550b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58b484aa42b2ddf8c6c9774a205269df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/037025699baf3ff5b27e02f2561b0d09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3be362dec96173f246ff747264007817.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f4ee9f2ac6e032f19afa201fd6f7480.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
您最近一年使用:0次
5 . 无穷数列
满足:
,且当
时有:
(表示最大项).
(1)若
,求
的所有可能值;
(2)若存在正整数T,对
,有
,证明:
是数列各项中的最大项;
(3)在(2)的条件下,
,试求m的所有取值的个数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9a27ff508b72a5b33614a010f73b457.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6241e69135558b636893a588ebffc1f6.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039e4fe671d61e59b96ee525c9df43e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
(2)若存在正整数T,对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71b78297a65e7fad69635b19928ecc10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b2b94cbf8f1acc77ed2618d9ba5756a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
(3)在(2)的条件下,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e0ccc1b3de66bc3a6bc666c4cb5ddb3.png)
您最近一年使用:0次
6 . 已知数列
的各项均为非负实数,且对任意正整数
,均有
.
(1)若
成等差数列,证明:存在无穷多个正整数
,使得
;
(2)若
,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/336407650f4fc2bdfdbe639c03ebb599.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c14d9ae06f864498048d55088ff4e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea867f78d52b7382c38e8fc6e2e2a9c0.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16c2570afdfeebd6bfbcc6da5edc878f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce88126c3cbc88e03d38f56b7da315b6.png)
您最近一年使用:0次
7 . 若
,
,求数列
的前n项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0d00dc95245a5cc7ed506bd0f50498e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b933d363f5b85a6b4110c9ef030c56a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2767882820f4ba0defde0e412adb747f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
您最近一年使用:0次
2023高三·全国·专题练习
8 . 已知数列
满足
,
.
(1)证明:
.
(2)设
为数列
的前n项和,证明:
.
![](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/78e887f2825f2f1cc2fab564b0bd35ec.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c46ca8b47107015953c8385c52a57f.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee77e52bf090220c5e8e440952b9607e.png)
您最近一年使用:0次
2023高三·全国·专题练习
9 . 设数列
满足
,
.
(1)证明:
.
(2)设数列
的前n项和为
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ea8d0e50065114b05ef2dc1ea1129cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb2dbc4c08aa70076c1c12daeedcb298.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c31f2f5b97bc76078c101082bb76bb6.png)
(2)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41cf1da18d91f7c98086553d157d1a87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ffb575d8e0365de6ccab0d0645fc78a.png)
您最近一年使用:0次