1 . 定义:对于任意一个有穷数列,第一次在其每相邻的两项间都插入这两项的和,得到的新数列称之为一阶和数列,如果在一阶和数列的基础上再在其相邻的两项间插入这两项的和称之为二阶和数列,以此类推可以得到n阶和数列,如
的一阶和数列是
,设它的n阶和数列各项和为
.
(1)试求
的二阶和数列各项和
与三阶和数列各项和
,并猜想
的通项公式(无需证明);
(2)若
,求
的前n项和
,并证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c77a27ecc68192b122861b8c4689ce29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35d0a8da5206f1114ead419f47b81044.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(1)试求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c77a27ecc68192b122861b8c4689ce29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6899bf9cadae2ccdb14cbc87d4f280ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7244499d5babf433375d0b71a672a927.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f640e13b3a3880bf49a49845eee47f07.png)
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2 . 费马数是以数学家费马命名的一组自然数,具有形式:
,
.1732年,数学家欧拉算出
不是质数,从而宣告费马数都是质数的猜想不成立.现设
,
,
为数列
的前n项和,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75734270b367c16d5621c4e3027c4ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30ac5abd893e2158c86f56e697f452ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cd73c5a7998f990819ff677357c469c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eab7f48841750dfed7f33761e6c9a725.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
A.![]() | B.![]() |
C.![]() | D.![]() ![]() |
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3 . “垛积术”在我国古代早期主要用于天文历法,后来用于求高阶等差级数的和.元代数学家朱世杰在沈括(北宋时期数学家)、杨辉(南宋时期数学家)研究成果的基础上,在《四元玉鉴》中利用了“三角垛”求一系列重要的高阶等差级数的和.例如,欲求数列
,
,
,…,
,
的和,可设计一个正立的
行三角数阵,即正三角形
的区域中所有数的分布规律为:第1行为1个
,第2行为2个
,第3行为3个
,…,第
行为
个1;再选一个数列
(其前
项和已知),可设计一个倒立的
行三角数阵,即正三角形
的区域中所有数的分布规律为:第1行为
个
,第2行为
个
,第3行为
个
,…,第
行为1个1.这两个三角数阵就组成一个
行
列的菱形数阵.若已知
,则运用垛积术,求得数列
,
,
,…,
,
的和为____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d8d28c824b791078bd9e60a636cebd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6780dd20734ecb6865a4ec9bae255b15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1025927646fd51373b385bb5ed9dceb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e94613fe8f9b0fc1ba68e541a1ddad6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56eb12f6196469a8d1e556cb0fab6085.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c0cdbf7b7cb42491810101c6e0db4ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a1271285428c740905f7d5db68c5dc6.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ab8a7e95d65fd5fcb3650297ec75a9c.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a0876215b2fd463d151523cd3c6b447.png)
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2023-05-23更新
|
969次组卷
|
7卷引用:热点04 数列求和及综合应用-2022年高考数学【热点·重点·难点】专练(全国通用)
(已下线)热点04 数列求和及综合应用-2022年高考数学【热点·重点·难点】专练(全国通用)(已下线)第三篇 数列、排列与组合 专题2 多边形数、伯努利数、斐波那契数、洛卡斯数、明安图数与卡塔兰数 微点2 多边形数综合训练(已下线)专题11 数列前n项和的求法 微点1 公式法求和(已下线)专题18 数列中的创新题的解法 微点2 数列中的创新题综合训练(已下线)模块三 失分陷阱2 不会从情境中抽出数列模型或关系贵州省盘州市2021届高三第一学期第一次模拟考试理科数学试题福建省莆田第五中学2023-2024学年高二下学期第一次月考数学试卷
名校
解题方法
4 . 在数字通信中,信号是由数字“0”和“1”组成的序列,“0,1数列”是每一项均为0或1的数列,设
是一个“0,1数列”,定义数列
为数列
中每个0都变为“
”,每个1都变为“
”所得到的新数列.例如数列
,则数列
.已知数列
,记数列
,则数列
的所有项之和为___________ ;数列
的所有项之和为___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2231c2f7241d4fb4c48c3105700ea0c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/823b15e6dc9fce202c3c57e7d18df0f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f32df58be46198b2e7a112ed255d8bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/021e539f6eb01379bb6bb178d7711bf7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/169315556239b6ba1e63a39999b1dc39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0386aa1cce19b412747b4459394658fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/247b6beb51e8a29ae55a22fa33c70cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab94459e87c666facddbe1a23ae1899d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcc31dcdb99754fc452ff2b92a2fb8c9.png)
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5 . 已知
为正整数数列,满足
.记
.定义A的伴随数列
如下:
①
;
②
,其中
.
(1)若数列A:4,3,2,1,直接写出相应的伴随数列
;
(2)当
时,若
,求证:
;
(3)当
时,若
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/281440c5e428da28c0a40fecbb87a83a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b97559b8ae5f9544c7b93bf2f9d03394.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6559598727fb120a5cdbf4f15510615d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c995ba5a9caa036977b023f57a4202f.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/271c5044aeaf0fd2a6f75746754565c8.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/880b1efd3798a3ccf2633252b10e0ab9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/570d7b5b193a644beb91889bbde27cde.png)
(1)若数列A:4,3,2,1,直接写出相应的伴随数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3053e2b8a6bbc35527a1e4505b84ed0f.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2578cb9428c41fa9236c6350bae49f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/941e10d4febad08273c2b181023f019f.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2578cb9428c41fa9236c6350bae49f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4fac29b7c846a7ba3b612b0f7ebee41.png)
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2023-01-12更新
|
955次组卷
|
4卷引用:北京市西城区2022届高三二模数学试题变式题16-21
6 . 角谷猜想又称冰雹猜想,是指任取一个正整数,如果它是奇数,就将它乘以3再加1;如果它是偶数,则将它除以2.反复进行上述两种运算,经过有限次步骤后,必进入循环圈
.如取正整数
,根据上述运算法则得出
,共需要经过8个步骤变成1(简称为8步“雹程”),已知数列
满足:
(m为正整数),
①若
,则使得
至少需要_______ 步雹程;②若
;则m所有可能取值的和为_______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a16f78ce0dab1ac8fa6abbd70f2b008.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3967d620e2fef3ecc724c66e29f68a8.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/999ac8c1ef39251e07a7fc54cbf7e26e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61d1f72117ae0005865805bc63595574.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8e6261372802c3eea7084aa892b26c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e15ffa7fecea3704dc892ea8cd513c59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/890ccb4e11f70158cab2f46137c69aac.png)
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2022-05-20更新
|
1964次组卷
|
4卷引用:专题04 数列的通项、求和及综合应用(精讲精练)-3
(已下线)专题04 数列的通项、求和及综合应用(精讲精练)-3(已下线)2023年四省联考变试题11-16河北省唐山市2022届高三三模数学试题北京市师大附中2022-2023学年高二上学期数学期末试题
名校
解题方法
7 . 若无穷数列
满足以下两个条件,则称该数列为
数列.
①
,当
时,
;
②若存在某一项
,则存在
,使得
(
且
).
(1)若
,写出所有
数列的前四项;
(2)若
,判断
数列是否为等差数列,请说明理由;
(3)在所有的
数列中,求满足
的
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f09757d013574cf058d5bb944fdf034a.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ff2d243ad2dc2094c8b3ea5672cfebd.png)
②若存在某一项
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9c3ab5a5109ec773eadecca155377a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/188bbb34acf76a5c5aa35c7faf9ef7d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92d37541605dfedeb0c28921950e362d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72ac49ab7c8001c209b8611b9ea40d85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24431f390ba671b4de0d6abaeb9cf476.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8d9b27f78829b57da918aa20936a198.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f09757d013574cf058d5bb944fdf034a.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b967232e28ad0d453adc66676bdf8b2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f09757d013574cf058d5bb944fdf034a.png)
(3)在所有的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f09757d013574cf058d5bb944fdf034a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce8cd9b2d7084a9db3df313891d64d9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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2023-03-18更新
|
982次组卷
|
8卷引用:专题12压轴题汇总(10、15、21题)
专题12压轴题汇总(10、15、21题)专题07数列北京卷专题18数列(解答题)(已下线)北京市第四中学2024届高三上学期10月月考数学试题变式题16-21北京市石景山区2023届高三一模数学试题北京市第三十五中学2022-2023学年高二下学期期中测试数学试题北京市人大附中石景山学校2024届高三上学期10月检测数学试题单元测试B卷——第四章 数列
8 . 设等比数列
满足
,
.
(1)求数列
的通项公式和
;
(2)如果数列
对任意的
,均满足
,则称
为“速增数列”.
(ⅰ)判断数列
是否为“速增数列”?说明理由;
(ⅱ)若数列
为“速增数列”,且任意项
,
,
,
,求正整数
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/463a4d5f2d1d893498d1be84167a747f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73f3eafa7d128eb70334f1b796fc043c.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88dcc8896d7404df42fc515b31f9fff3.png)
(2)如果数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20be95c3038f55e438c0fb7598d6329b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(ⅰ)判断数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(ⅱ)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33f3c2d176cee47c5f0f33a69130a910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59dd6c97d2ee3e74ba5730f1cbcc1d43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/627e48c5ab76f5d1874c57a40d32d89e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fd4cde52680aee9a0e8b28568e91ee4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
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名校
解题方法
9 . 任取一个正整数,若是奇数,就将该数乘3再加上1;若是偶数,就将该数除以2.反复进行上述两种运算,经过有限次步骤后,必进入循环圈
.这就是数学史上著名的“冰雹猜想”(又称“角谷猜想”).如取正整数
,根据上述运算法则得出
,共需经过8个步骤变成1(简称为8步“雹程”).现给出冰雹猜想的递推关系如下:已知数列
满足:
(
为正整数),
当
时,
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a16f78ce0dab1ac8fa6abbd70f2b008.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3967d620e2fef3ecc724c66e29f68a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a73700b5135fc6a9c2d923a27a4c9b36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/999ac8c1ef39251e07a7fc54cbf7e26e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d4914df4e75585d5ff7709d64a23611.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8a3cc8c48bf54ec8252e5dce6867754.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59097ad7c8f3fcff871ad48933d30498.png)
A.170 | B.168 | C.130 | D.172 |
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2024-01-12更新
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922次组卷
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4卷引用:信息必刷卷04(天津专用)
10 . 已知数列
的各项是奇数,且
是正整数
的最大奇因数,
.
(1)求
的值;
(2)求
的值;
(3)求数列
的通项公式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97daaae5be89c76c7ccb25fd96339b46.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ab24d03d347b53c928e704601e68a7d.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcf4e20ea341827ce5f9552daee39462.png)
(3)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/846fa57d92d6ad44d6a0cafad1e71ed4.png)
您最近一年使用:0次
2024-05-08更新
|
1018次组卷
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3卷引用:5.2 等差数列和等比数列(高考真题素材之十年高考)