若数列
若满足递推关系
其中
为常数,我们称该数列为k阶常系数齐次线性递推数列,并称方程
为递推关系式(*)的特征方程,该方程的根称为数列
的特征根.我们有以下结论:对于k阶常系数齐次线性递推数列,若其不同的特征根为
,
,…,
,且特征根
的重数为
,则数列
的通项公式为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a573c3e3b4c5b02a85d309aee9ffbc2.png)
其中
,
,这里
都是常数,它们由数列初始值可以确定.
(1)若数列
满足
,且
,
,
,求数列
的通项公式;
(2)若数列
满足对于所有非负整数m,n(
),
都成立,且
,求数列
的通项公式;
(3)设边长为1的正六边形ABCDEF,O是六边形的中心,除了六边形的每一条边,我们还从点O到每个顶点连一条线段,共得到12条长度为1的线段,一条路径是指动点沿着上述线段(全部或部分)移动,始点终点均为点O的一条移动路线.求长度为2024的路径共有多少条?(注:根的重数就是方程中同样根的数量)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22ab67ba8b0719104e78cfa6ce029290.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11e26bb035fe18631ca09dd61ba446d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0c87ab1d7f0eaf58fb90e7087ad7e71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8256967311eda335e21bb88f6e726fb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec6ba141730fd5aae78ada1a8eb17d21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d68e94f023b09352f46cf2ff3afb291c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a573c3e3b4c5b02a85d309aee9ffbc2.png)
其中
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed9176aeda3df453783774182340e074.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f9b96ef08d0169c0c8ff9a06eb0c5d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1c5521a39235f0b9cdf432d5903aa83.png)
(1)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac06043337b08fece3c5762766fdb2a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/329785900390130a04a57d0b55aaa569.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ddb84ee3769b8977d138638120ed820.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bd68dad20a530c17474ad6c73be07e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b67fc21d26aead8dcbfb36d7df8aa895.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13a6e1d671215fc96e4bee3541d1096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)设边长为1的正六边形ABCDEF,O是六边形的中心,除了六边形的每一条边,我们还从点O到每个顶点连一条线段,共得到12条长度为1的线段,一条路径是指动点沿着上述线段(全部或部分)移动,始点终点均为点O的一条移动路线.求长度为2024的路径共有多少条?(注:根的重数就是方程中同样根的数量)
更新时间:2024-05-25 22:10:14
|
【知识点】 数列新定义
相似题推荐
解答题-问答题
|
困难
(0.15)
解题方法
【推荐1】若数列
对任意连续三项
,均有
,则称该数列为“跳跃数列”.
(1)判断下列两个数列是否是跳跃数列:
①等差数列:
;
②等比数列:
;
(2)若数列
满足对任何正整数
,均有![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc47df306e4bc72632047fbe01619bf9.png)
.证明:数列
是跳跃数列的充分必要条件是
.
(3)跳跃数列
满足对任意正整数
均有
,求首项
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4993aaab02cbc3cbed15d025f4b4e6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69dc2d44dfdb07c42acfb6eb6eec1f69.png)
(1)判断下列两个数列是否是跳跃数列:
①等差数列:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbc28deef24c776e671639e6cfc028fa.png)
②等比数列:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73a9819764e7f56202270fd85e6841c9.png)
(2)若数列
![](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/bc47df306e4bc72632047fbe01619bf9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff03fcfa701aa0f42e914bd82afaaeb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1381f0937c6052ce088e0eaee7df4880.png)
(3)跳跃数列
![](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/2ae8a1b864e3e6d37a0eb027e661d9ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
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解答题-证明题
|
困难
(0.15)
解题方法
【推荐2】记无穷数列
的前
项中最大值为
,最小值为
,令
,则称
是
“极差数列”.
(1)若
,求
的前
项和;
(2)证明:
的“极差数列”仍是
;
(3)求证:若数列
是等差数列,则数列
也是等差数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ddad3d9fdb5e9951b6a1c31f9a72a71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0a625b91e0eba33b107550ee2a1e2f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07502f5de77ea134859dbfd235b3ee23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43b7e7cd571c8cd141cbbfe5d0890bf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c9d38a1171131b1a1f3f70ca2117be1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43b7e7cd571c8cd141cbbfe5d0890bf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43b7e7cd571c8cd141cbbfe5d0890bf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43b7e7cd571c8cd141cbbfe5d0890bf6.png)
(3)求证:若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43b7e7cd571c8cd141cbbfe5d0890bf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
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困难
(0.15)
名校
解题方法
【推荐3】大数据环境下数据量积累巨大并且结构复杂,要想分析出海量数据所蕴含的价值,数据筛选在整个数据处理流程中处于至关重要的地位,合适的算法就会起到事半功倍的效果.现有一个“数据漏斗”软件,其功能为;通过操作
删去一个无穷非减正整数数列中除以M余数为N的项,并将剩下的项按原来的位置排好形成一个新的无穷非减正整数数列.设数列
的通项公式
,
,通过“数据漏斗”软件对数列
进行
操作后得到
,设
前n项和为
.
(1)求
;
(2)是否存在不同的实数
,使得
,
,
成等差数列?若存在,求出所有的
;若不存在,说明理由;
(3)若
,
,对数列
进行
操作得到
,将数列
中下标除以4余数为0,1的项删掉,剩下的项按从小到大排列后得到
,再将
的每一项都加上自身项数,最终得到
,证明:每个大于1的奇平方数都是
中相邻两项的和.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e402188dc9ca4cf144aae335d6b2481a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbb2f1272b626e0fd5c79c1a91d48b07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5be101dbc74d4270def39679782f166b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039b824cebbc6b9c96efbf2c7ae6372a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec8b68d90e55b00b6662a23f851cca89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(2)是否存在不同的实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4c8b61dd87a6a1edecdc58b11714b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0833aa85a3389c7fc576b5f55359100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6996aacf881b439908670c81a749ddd5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a45163ab6dd6f8ca9f21cabfebcbe4cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4baee6e40cd05d5831ff42045f14cd97.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/992690ef9791e38b2ef8b3123b7bdae1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5be101dbc74d4270def39679782f166b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78fb02daaf8ffbb23ec24c66a7483ccd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/344b314161c9cdaf34cc8dbe9c3adb1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/089371cf5f29c549977e2f4b8dd8b8ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/089371cf5f29c549977e2f4b8dd8b8ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/568ef2909bf624c3d346474741d226b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/568ef2909bf624c3d346474741d226b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5ab0309e2cd35585ea9fb2cc3017abf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5ab0309e2cd35585ea9fb2cc3017abf.png)
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