解题方法
1 . 卷积运算在图象处理、人工智能、通信系统等领域有广泛的应用.一般地,对无穷数列
,
,定义无穷数列
,记作
,称为
与
的卷积.卷积运算有如图所示的直观含义,即
中的项依次为所列数阵从左上角开始各条对角线上元素的和,易知有交换律
.
,
,
,求
,
,
,
;
(2)对
,定义
如下:①当
时,
;②当
时,
为满足通项
的数列
,即将
的每一项向后平移
项,前
项都取为0.试找到数列
,使得
;
(3)若
,
,证明:当
时,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e30e288d8e1764fde9cf193e2bdc204.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bf62b0e90a1b3340c00e2a2e1f8ac85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5ab0309e2cd35585ea9fb2cc3017abf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dbbee347cb4a4a531e83a545c5cf306.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b80c1ed7b10ac7ca1cd81cdd39a8fcc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/267c99ff3f6386113dbaa7b1e49612da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bf62b0e90a1b3340c00e2a2e1f8ac85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7936359df4c926b72b48c6fdae55f12d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b76f79be89b8c6227b68eded6b675546.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db84454f051d418a4904fa423ab8b304.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6fdd7e9cd5d1764bc5de8add15700ae.png)
(2)对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f79f03f7df0441257dec04bd4ceb0ae2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b019ff582c6b6f95dc19b4bd4ced254a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c45176df950dfe48b8ca7eac08ee349.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ca21a96fc2f9f9fab356f81599fc01c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1af554aa9625a1c75ad96d9bc3b6c392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b019ff582c6b6f95dc19b4bd4ced254a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4945771db04cd575d385e9355e8a0829.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a77316e06c00a9086be642f7f590684.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/711ca723e111e4db0a70ce4c47ee66d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/711ca723e111e4db0a70ce4c47ee66d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8016ef5b66902b71162f20fa0ed02c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6933f66320908d7232fdc92e31bb6123.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b80c1ed7b10ac7ca1cd81cdd39a8fcc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bf62b0e90a1b3340c00e2a2e1f8ac85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd3517fe4fa5b7eeb4878d35ee94a35e.png)
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名校
解题方法
2 . 斜二测画法是一种常用的工程制图方法,在已知图形中平行于
轴的线段,在直观图画成平行于
轴(由
轴顺时针旋转
得到)的线段,且长度为原来的
,平行于
轴的线段不变.如图,在直角坐标系
中,正方形
的边长为
.定义如下图像变换:
表示“将图形用斜二测画法变形后放回原直角坐标系”;
表示“将图形的横坐标保持不变,纵坐标拉伸为原来的
倍”.
经过两次
变换后所得图形为
,求
的坐标;
(2)在第
次复合变换中,将图形先进行一次
变换,再进行一次
变换,
. 记正方形
进行
次复合变换后所得图形为
.过
作
的垂线,垂足为
,若
恒成立,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b4d2174f411d9db6ab7b2aea47818cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1e5fa72f2878b476bc57f0df12d6555.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e9a724b59c890095baa5cb73e267c44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d31c9ff64b11c29441ffc10c8cc70cba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89fe33c85f43cc3208ae16c2796b9188.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e9a724b59c890095baa5cb73e267c44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5bf350a619ef25d8d9b988f3db804e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68ee712dfc82e1acc31ef8dcad479a39.png)
(2)在第
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e9a724b59c890095baa5cb73e267c44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d31c9ff64b11c29441ffc10c8cc70cba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d904903ab8465eb522d2b8cde0fc29a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36134f01da0f13b340e82e8835324f25.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3cfeacc29e6a61c5b3b4e439c0a91df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f24172ca004ead2629ef8541a709419.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87bd7d18f67e90a7c37fad4252e43c9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a8c8bb5b1ee645a5e94c72823b5f295.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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2024-05-08更新
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816次组卷
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2卷引用:浙江省杭州学军中学2024届高三下学期4月适应性测试数学试题
名校
解题方法
3 . 已知
是方程
的两根,数列
满足
,
,
.
满足
,其中
. 则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d98919b1335a8b7ca020636d1494ad0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/073c95775e8c6c15c7f2f8a4a2ad050b.png)
![](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/19a31ae70f96dc4aef6e1ca3ef9fed38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf6746bfcdf694447215a11f5b677d40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/140b5428dace7d41a3967db2f60d633e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c071ed6bc646439b162e096e64fbcd50.png)
A.![]() |
B.![]() |
C.存在实数![]() ![]() ![]() |
D.不存在实数![]() ![]() ![]() |
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2024-05-08更新
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946次组卷
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2卷引用:浙江省杭州学军中学2024届高三下学期4月适应性测试数学试题
名校
4 . 甲口袋中装有2个黑球和1个白球,乙口袋中装有1个黑球和2个白球.设从甲、乙两个口袋中各任取一个球交换放入另一个口袋为一次操作,经过
次这样的操作,记甲口袋中黑球个数为
.
(1)写出
的分布列并计算
;
(2)某人重复进行了100次操作,记
,
,求该数列
的前100项和
的最大值;
(3)定性分析当交换次数趋向于无穷时,
趋向的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93d0f3799612b81e85b87241ec8eee68.png)
(1)写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/096b1ece1dcd29c59a46a4b3e02cb548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4288f3086ea23cd82eff43f46f5c5a5.png)
(2)某人重复进行了100次操作,记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d8edc0a00a2c28cd69e2c50a27022fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3976a94e7409ea3fa3ee76b73493eae0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08c9965a04c2a6de04e949a15762f372.png)
(3)定性分析当交换次数趋向于无穷时,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/229494e1240a594592035d23283fedbc.png)
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解题方法
5 . 数列
满足
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e944e87fd2674c579e0595745bc67b9b.png)
A.当![]() ![]() ![]() ![]() |
B.当![]() ![]() ![]() ![]() |
C.当![]() ![]() ![]() ![]() |
D.当![]() ![]() ![]() ![]() |
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6 . 数列
满足
,
,则
( )
![](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/9c4dd3824d6d40c0a8927f655f0c77c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24360d9f7f45dc37134ac9490926225e.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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解题方法
7 . 若
,且对任意正整数n,均有
,则称一个复数数列
为“有趣的”.若存在常数C,使得对一切有趣的数列
及任意正整数m,均有
,则C的最大值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b670aca396b96eaf2c553b1ca84486dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a383b03b4869ea984d58b8d87c35402.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/078c417ea54a5065c1f72941b9e4b0be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/078c417ea54a5065c1f72941b9e4b0be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7763a0445ba78bceb1bc24a9f1010f5c.png)
A.![]() | B.1 | C.![]() | D.![]() |
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4卷引用:浙江省杭州市桐庐中学2022-2023学年高三上学期1月期末数学试题
浙江省杭州市桐庐中学2022-2023学年高三上学期1月期末数学试题(已下线)专题17 数列探索型、存在型问题的解法 微点1 数列探索型问题的解法河南省新乡市第二中学2024届高三上学期1月测试数学试题(已下线)【练】专题3 数列范围(最值)问题
名校
8 . 已知数列
中,
,
,则
等于
( )
![](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/420f8aefb8261d70bf8cb27560a0ec3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90707999e8fc89ae1137e5115c39f637.png)
![](https://img.xkw.com/dksih/QBM/2023/10/19/3349545285345280/3350776202764288/STEM/f5f07eb7627e452b95b0d3bf53e8bdc9.png?resizew=4)
A.![]() | B.![]() | C.![]() | D.![]() |
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18卷引用:浙江省杭州高级中学2021-2022学年高二上学期期末数学试题
浙江省杭州高级中学2021-2022学年高二上学期期末数学试题江苏省苏州市常熟市2020-2021学年高二上学期期中数学试题山东省泰安市肥城市2020-2021学年高二上学期期末数学试题山东省泰安市宁阳县第一中学2020-2021学年高二上学期期末数学试题山东省泰安市2020-2021学年高二上学期期末数学试题江苏省淮安市淮安区2020-2021学年高二上学期期中数学试题河北省实验中学2022届高三上学期开学考试数学试题(已下线)第01讲 数列-【帮课堂】2021-2022学年高二数学同步精品讲义(苏教版2019选择性必修第一册)江苏省苏州市张家港市2021-2022学年高二上学期期中数学试题黑龙江省大庆市让胡路区大庆中学2021-2022学年高二上学期期末数学试题安徽省六安外国语高级中学2021-2022学年高二上学期期末数学试题安徽省滁州市定远县育才学校2021-2022学年高二分层班下学期期中考试数学(文)试题江苏省苏州市常熟外国语学校2023-2024学年高二上学期10月月考数学试题(已下线)模块三 专题5 数列中复杂递推式问题(高三人教A)四川省成都金苹果锦城第一中学2023-2024学年高二上学期期中考试数学试题山东省青岛市第十七中学2022-2023学年高二上学期期末数学试题(已下线)高二数学第一学期期期末押题密卷06卷(已下线)专题32 数列的概念及性质7种常见考法归类 - 【考点通关】2023-2024学年高二数学高频考点与解题策略(人教A版2019选择性必修第二册)
名校
解题方法
9 . 已知函数
,数列
的前
项和为
,且满足
,则下列有关数列
的叙述正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/307bb5726473ed17fc82d14215f435f4.png)
![](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/a27342dd38781f697127de9142861639.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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10 . 已知数列
满足:
,且
,下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9645bd4d2002993b90ec6d48f9c04f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9598459d3ad5d1cab0f50b5f9bcd2562.png)
A.若![]() ![]() | B.若![]() ![]() |
C.![]() | D.![]() |
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8卷引用:浙江省杭州市高级中学2020届高三下学期教学质量检测数学试题
浙江省杭州市高级中学2020届高三下学期教学质量检测数学试题浙江省杭州高中2020届高三下学期5月高考质检数学试题(已下线)4.1 数列-2021-2022学年高二数学同步培优训练系列(苏教版2019选择性必修第一册)(已下线)第四章 数列(提分小卷)-【单元测试】2021-2022学年高二数学尖子生选拔卷(苏教版2019选择性必修第一册)(已下线)专题19 数列的综合应用-3(已下线)第三篇 数列、排列与组合 专题5 迭代数列与极限 微点5 迭代数列与蛛网图(已下线)重难点突破02 函数的综合应用(九大题型)(已下线)重难点突破01 数列的综合应用 (十三大题型)-2