1 . 对于正整数m,n,存在唯一的自然数a,b,使得
,其中
,我们记
.对任意正整数
,定义
的生成数列为
,其中![](https://staticzujuan.xkw.com/quesimg/Upload/formula/874689ac8a6f70ecd93aa1aaf7d626f6.png)
.
(1)求
和
.
(2)求
的前3项.
(3)存在
,使得
,且对任意
成立.考虑
的值:当
时,定义数列
的变换数列
的通项公式为
当
时,定义数列
的变换数列
的通项公式为
若数列
和数列
相同,则定义函数
,其中函数
的定义域为正整数集.
(ⅰ)求证:函数
是增函数.
(ⅱ)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bad12b60594a6408ccc237cad2880088.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ff38a5b95f1fb452d87fce9c80b1249.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/574aa06d3d4f9952157c60ec8f40b839.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec5914cd027a87542e12cfb0c92c0aa1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/874689ac8a6f70ecd93aa1aaf7d626f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57dbb663375b927f3c00d1be3fda1508.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37dbd8745cf6246eea337a1bf91258c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c87dc8de41005802093ae66c08f46a7a.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/926145a2b2232a88ceed6e69dc050265.png)
(3)存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d7e9f86738335a22298559db41037a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/623ebc0f0eba98e8f63b7cd982c11009.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8144524289a76dadf88c02976acea2db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebf0ce448d146abfcd0bd689b925d168.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c20d94765a05ee86edf370dd64fcff7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec5914cd027a87542e12cfb0c92c0aa1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d054e320d843bf0ca4a620519f75ebaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8121a889da1f3ecbecd5387a9473c0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d98b98330b62847c220ab2f127e11391.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec5914cd027a87542e12cfb0c92c0aa1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d054e320d843bf0ca4a620519f75ebaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/caf19212dd24556423636ce37034cdf7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d054e320d843bf0ca4a620519f75ebaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96dbc1106da9238ec6300e13ca3d9b10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46498efb2e485c1172abe2189e498613.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b88ba8e1dc91318d8a9e1856d8ae080e.png)
(ⅰ)求证:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b88ba8e1dc91318d8a9e1856d8ae080e.png)
(ⅱ)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bee5e2ada7f1e99bf68508d21b8f6fb7.png)
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2024高三下·全国·专题练习
2 . 由
,
给出的数列是著名的斐波那契数列:
,其中每一个数均称为斐波那契数.则斐波那契数列中_________ 末尾是三个0的斐波那契数.(填存在或不存在)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64e6dcb540f7df224600858f30d72efa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/831909d320b5325eeb5642a8ce1874af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66b7a7181ddbdb7499c014a4c4760016.png)
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3 . 已知数集
具有性质P:对任意的
,使得
成立.
(1)分别判断数集
与
是否具有性质P,并说明理由;
(2)已知
,求证:
;
(3)若
,求数集A中所有元素的和的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30b7edfe73bdd0bb2a6e84512b62bdc7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a921d157d198de0f934da07e16dc7df3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a1e76d1341e8e6bd89b7075150536bd.png)
(1)分别判断数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59970351aa04d29f62d480c7280763e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37a3069c8accda13019e775a5dc198c2.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/108bce68aab5565c4ed9a0c3e11150e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7006f52b1f7cf1bdf8374bd2da3e4562.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d91a94ec87afbc073e077f2c453a304b.png)
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2022-05-13更新
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1003次组卷
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7卷引用:北京市房山区2022届高三二模数学试题
2021·全国·模拟预测
名校
解题方法
4 . 已知数列
满足
,
,其中
表示不超过实数
的最大整数,则下列说法正确的是( )
![](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/9a4e126ed47f61090b830d2d02340386.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
A.存在![]() ![]() | B.![]() |
C.![]() | D.![]() |
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名校
解题方法
5 . 对于正整数n,设
是关于x的方程
的实数根.记
,其中
表示不超过x的最大整数,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7999465d0e871febde66296a0cbf058c.png)
____________ ;设数列
的前n项和为
则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/469a49d15c634c79d8e36cb6edaff624.png)
___ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2147cdb7c47327b9a8e3274df089aec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c84fb3e536f74961243c6b89ddcee09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7999465d0e871febde66296a0cbf058c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/469a49d15c634c79d8e36cb6edaff624.png)
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2020-08-12更新
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1961次组卷
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6卷引用:湖北省七市州教科研协作体2020届高三下学期5月联合考试数学(理)试题
6 . 已知
,数列
中的每一项均在集合
中,且任意两项不相等,又对于任意的整数
,均有
.记所有满足条件的数列
的个数为
.例如
时,满足条件的数列
为1,2或2,1,所以
.
(1)求
;
(2)求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/263faa85998ecb4857b97f40a9fa6e03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b28c42df33e1a5fe54b412acb789879.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b739e2a6594f3b0c3b5099750062a96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d28bc1b09a0c1a3fecac62a86c93b85c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c8876e4e9512be5fbfe5fe4e4fb7527.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc2d3df37e73a8abea815f37dbb3fff5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73e96fafcc7b7f783d436f853449208d.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57483e04fd1840c87ac5325157149877.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
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7 . 定义
为正整数
的各位数字中不同数字的个数,例如
.在等差数列
中,
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3f7fda69e2b32b9ced2239f915fa59b.png)
___________ ,数列
的前100项和为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/006d54905e72dac6b9f6fcf1a6ada6d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66ebe46bffead42216839a2b4958a420.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/903304fb655d626931c005a4dbc8ea29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3f7fda69e2b32b9ced2239f915fa59b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65723ad260705c559714d9b12e20a1fd.png)
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2020-05-09更新
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945次组卷
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8卷引用:2020届陕西省商洛市高三下学期高考模拟测试文科数学试题