1 . 斐波那契数列又称为黄金分割数列,在现代物理、化学等领域都有应用,斐波那契数列
满足
,
.给出下列四个结论:
①存在
,使得
成等差数列;
②存在
,使得
成等比数列;
③存在常数t,使得对任意
,都有
成等差数列;
④存在正整数
,且
,使得
.
其中所有正确结论的序号是________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8323901a49cac29afd7d62864f088077.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ea81c176437113bfdc27362aacd5dad.png)
①存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24f8494594299d0ecce6e1e52151f402.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a91239c38be30570f5905f56d03b0ecb.png)
②存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24f8494594299d0ecce6e1e52151f402.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a91239c38be30570f5905f56d03b0ecb.png)
③存在常数t,使得对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15a70b95c53fb6655721e2a8c61f5c2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed75c7d0e5b35f5faa57cdc09c8a134a.png)
④存在正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a8eaeeab1ff32f8f15696eb18fdc0e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0844d2b5218031f4a67807468b02653c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a00eb8a57a82e7c87e85c575677e3d26.png)
其中所有正确结论的序号是
您最近一年使用:0次
2023-05-05更新
|
1577次组卷
|
6卷引用:北京卷专题17数列(填空题)
北京卷专题17数列(填空题)北京市朝阳区2023届高三二模数学试题(已下线)专题11 数列前n项和的求法 微点9 转化化归法求和(已下线)等差数列与等比数列(已下线)【讲】 专题8 斐波那契数列上海市普陀区2024届高三上学期期中调研测试数学试题
名校
解题方法
2 . 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)
您最近一年使用:0次
2023-03-18更新
|
971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)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(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单元测试B卷——第四章 数列
解题方法
3 . 对非空数集
,
,定义
与
的和集
.对任意有限集
,记
为集合
中元素的个数.
(1)若集合
,
,写出集合
与
;
(2)若集合
满足
,
,且
,求证:数列
,
,
,
是等差数列;
(3)设集合
满足
,
,且
,集合
(
,
),求证:存在集合
满足
且
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a829fdd8ec0f3b7ede883cf2c3e53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a829fdd8ec0f3b7ede883cf2c3e53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d2879c154a00d481e9b73a3fd1c56f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc8279d9dd0b7750953cb9e2098b3b90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(1)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7c14c3a132f7fed19e8298890692bdd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb3f7bcd226dc1daa07500ffa9b57965.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ddbaf050f74e3b87e6d2f9647fc43fd3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec9f6ea6346066054b5c722763d6b026.png)
(2)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7c51a72af8fefb52643a2ccfe2c0100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea5867fde790c54e6a931c5d1d22b049.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c81439ea75624aba008bd0f6a35dbf15.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/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
(3)设集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7c51a72af8fefb52643a2ccfe2c0100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea5867fde790c54e6a931c5d1d22b049.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f1863bbba4b8653b4c99ff67fc3edb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eef61be75060a7d37c8bc03782be6530.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72ac49ab7c8001c209b8611b9ea40d85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0588c80fa0ee2598f12eb7725c2e406.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c822b25dca0587cea9e29c48849de7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a6ce6a51538d70382465e76b6d16f52.png)
您最近一年使用:0次
2022-03-30更新
|
1758次组卷
|
4卷引用:北京卷专题18数列(解答题)
4 . 设数集
满足:①任意
,有
;②任意
、
,有
或
,则称数集
具有性质
.
(1)判断数集
是否具有性质
,并说明理由;
(2)若数集
且
具有性质
.
(i)当
时,求证:
、
、
、
是等差数列;
(ii)当
、
、
、
不是等差数列时,写出
的最大值.(结论不需要证明)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c8647b00cc8c8f35555c7d78cf2812c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e2e79843faf62dde86bf858d1e0569.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45c8b2714e2f6ddfdd6b05d3b4de1149.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b8e5872f45d4b878b0119997cb5bae2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84734fbba70c0b45045fabf8090f810b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(1)判断数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2161a642b95463642adc3892850bc74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(2)若数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ddc189d2c675b0e2ade4f7ed40f66fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8509c4b8fef1e10a20fe1c3e9243ac8e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(i)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76c7867969b14fd642147188b6ebf29c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
(ii)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
您最近一年使用:0次
2021-09-26更新
|
584次组卷
|
7卷引用:北京卷专题18数列(解答题)
5 . 对于数列
,记
,其中
表示
这
个数中最大的数,并称数列
是
的“控制数列”,如数列
的“控制数列”是
.
(1)若各项均为正整数的数列
的“控制数列”为
,写出所有的
;
(2)设![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa7dedbf326725b8f06e6d64357e4032.png)
.
(i)当
时,证明:存在正整数
,使得
是等差数列;
(ii)当
时,求
的值(结果可含
).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e98d86863421802401afe64cf356bd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8424af9108fc00fbf86a3d5c9409e47.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41dd42e4f493477fb0f36137893d4d06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2d783f3b03d31465cd9314984162558.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96785775d4386fe51da91be3cc7b8630.png)
(1)若各项均为正整数的数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/933e22f83657cd917192e86daf0d6b12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa7dedbf326725b8f06e6d64357e4032.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69d307ec71820b6536453fbdb5069da3.png)
(i)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e48d3a72a4f272e2bb9eb955680bfdf7.png)
(ii)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10dd628a48cf11a09a49d38b40d1ce26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/886306bda030997fb559c75e457af164.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
您最近一年使用:0次
6 . 数列
:
,
,
,…,
,…,对于给定的
(
,
),记满足不等式:
(
,
)的
构成的集合为
.
(Ⅰ)若数列
,写出集合
;
(Ⅱ)如果
(
,
)均为相同的单元素集合,求证:数列
,
,…,
,…为等差数列;
(Ⅲ)如果
(
,
)为单元素集合,那么数列
,
,…,
,…还是等差数列吗?如果是等差数列,请给出证明;如果不是等差数列,请给出反例.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.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/291c25fc6a69d6d0ccfb8d839b9b4462.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/544530e1133b2924ccfbe691141a5641.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60939f5f5cd85a28dcb63d2f78d26b60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2eb18547717a019d4b546b8dd0b0365c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/030137376417efb2ac10443ff54fbfb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13c0b3d5b60308da39aaf5493d58f444.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe779a0f086e93f260a1b0c9be9cc415.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e2ea3d72f46edfb7216c7bc9ab9cf9a.png)
(Ⅰ)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f41c721ebc7a5f8346da3c44af85a047.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7e612765b49f8cdda75bdaaf4f86edd.png)
(Ⅱ)如果
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e2ea3d72f46edfb7216c7bc9ab9cf9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60939f5f5cd85a28dcb63d2f78d26b60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/544530e1133b2924ccfbe691141a5641.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/3282e5fde4ae53fcb1bb072a685304c9.png)
(Ⅲ)如果
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e2ea3d72f46edfb7216c7bc9ab9cf9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60939f5f5cd85a28dcb63d2f78d26b60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/544530e1133b2924ccfbe691141a5641.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/3282e5fde4ae53fcb1bb072a685304c9.png)
您最近一年使用:0次
名校
解题方法
7 . 如图,曲线y2=x(y≥0)上的点P1与x轴的正半轴上的点Qi及原点O构成一系列正三角形,△OP1Q1,△Q1P2Q2,…,△Qn﹣1PnQn…设正三角形Qn﹣1PnQn的边长为an,n∈N*(记Q0为O),Qn(Sn,0).数列{an}的通项公式an=_____ .
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/25/befe99df-b658-421c-b586-e7753d0d746d.png?resizew=239)
您最近一年使用:0次
2020-03-25更新
|
2226次组卷
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12卷引用:专题7.18 数列与解析几何的综合-2022届高三数学一轮复习精讲精练
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真题
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8 . 设
和
是两个等差数列,记![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fae7c25bbcda4893fd243d929c01f969.png)
,
其中
表示
这
个数中最大的数.
(Ⅰ)若
,
,求
的值,并证明
是等差数列;
(Ⅱ)证明:或者对任意正数
,存在正整数
,当
时,
;或者存在正整数
,使得
是等差数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fce83115a50f99e08e9a2db7267aeed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fae7c25bbcda4893fd243d929c01f969.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d9812dcbb57996f2212b037918ab195.png)
其中
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b125c9321c0d8bd9cf942d6da8bebf16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36b14e03f30c56d9943e4a82d0e029b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5873c01192b7d33b7483f444f90b5b0.png)
(Ⅰ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b80c1ed7b10ac7ca1cd81cdd39a8fcc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15ff259bff098430a6512d0e4f6fb2d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/312893147a40a4cd5d46fc2ad309c488.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38ef4c4439b36c2847b0056a116d56d4.png)
(Ⅱ)证明:或者对任意正数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/856b137a34d2d5b20671b7a3c7a29606.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c738db07e589f0345db84933cfcb189.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7730387952855f771c18cf0bbf423be.png)
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