1 . 设
为给定的正整数,实数
及
满足如下条件:
(1)
;
(2)
;
(3)
;
(4)
.
证明:对一切
,均有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e3d172f08313520e76b6cbc2ff9980c.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a81357dc2bcd382142e2aa19a74cadd.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3785965a2dd765765ccef0330052fca7.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b04f7ed829546d2b2260985f507f3a8.png)
(4)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbe0bce8fb16093bc80ad1e0a472acf.png)
证明:对一切
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ef835c9ad2636a9662fb6c99e3abc78.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd7a172e3de92f315198a515eef6ebbe.png)
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2 . 对于正整数
,如果严格递增的非负整数数列
,
使得所有非负整数可以唯一地表示为
,其中i、j、k可以相同,则称数列
,
为
好的.
(1)证明:对任意正整数n,存在唯一的
好的数列.
(2)已知存在最小的正奇数m,使得在
好的数列中有
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ac19e2a797cd0a408316988a63b3755.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3a52572fcfb82b32f0d467b0e5e3418.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/490111a58b3debc2b0ebcac2481124a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3a52572fcfb82b32f0d467b0e5e3418.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c8559db5cec89fb0ed29e8be8fdb0b1.png)
(1)证明:对任意正整数n,存在唯一的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c8559db5cec89fb0ed29e8be8fdb0b1.png)
(2)已知存在最小的正奇数m,使得在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c8559db5cec89fb0ed29e8be8fdb0b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf2969976f3f125037d1384aa843b7cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1019d4ad2e3fb4a7abb66e0e9e55b556.png)
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3 . 若数列
,求证:存在无穷多个正整数n,使得
,并确定是否存在无穷多个正整数n使得
?(这里
表示不超过x的最大整数)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3f8dec8428b043470e96c94f99896a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2798e1dcab1f7f0fe3b8a94b3cd6a88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1254070260067f8bf2fec39a7d0c8f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
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4 . 对于数列
,若存在常数
使得
对任意正整数
成立,则称
是有界数列.已知数列
满足递推式
,求证:
(1)若
,则
不是有界数列.
(2)若
,则
是有界数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2480f87a11c4cd450bc9454ea7276722.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fbb510a00c96650e85fa03afed69d85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32bcb64530e6818698cc3d83c87a93c1.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13a6e1d671215fc96e4bee3541d1096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039e4fe671d61e59b96ee525c9df43e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
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解题方法
5 . 已知数列
满足:
,
.
(Ⅰ)证明:数列
为等比数列,并求数列
的通项公式;
(Ⅱ)记
,求使
成立的最大正整数n的值.(其中,符号
表示不超过x的最大整数)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2438f2272d7b7ab51dbbe587025a553d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5af8e317162f3c1bb3483b08207ea13.png)
(Ⅰ)证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/213e22890204937a5dded4436369390f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(Ⅱ)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b484a6f707521fb604b8139753d2a6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eeed2dd4e7c90200f05009bd071b3801.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
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2021-03-02更新
|
2043次组卷
|
7卷引用:浙江省名校协作体2021届高三下学期联考数学试题
浙江省名校协作体2021届高三下学期联考数学试题(已下线)精做02 数列-备战2021年高考数学(文)大题精做(已下线)精做02 数列-备战2021年高考数学(理)大题精做(已下线)专题20 数列综合-2020年高考数学母题题源全揭秘(浙江专版)(已下线)【新东方】高中数学20210429—010【2021】【高三下】(已下线)专题4.3 等比数列-2020-2021学年高二数学同步培优专练(人教A版2019选择性必修第二册)(已下线)第17节 等比数列及前n项和
6 . 已知正项数列{an}的前n项和为Sn,且
.
(1)求数列{an}的通项公式;
(2)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0d39ec2b059c1d625f22be212d73086.png)
(1)求数列{an}的通项公式;
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb16854fd92b514c746fbccda164d2f5.png)
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7 . 若正整数
的二进制表示是
,这里
(
),称有穷数列1,
,
,
,
为
的生成数列,设
是一个给定的实数,称
为
的生成数.
(1)求
的生成数列的项数;
(2)求由
的生成数列
,
,
,
的前
项的和
(用
、
表示);
(3)若实数
满足
,证明:存在无穷多个正整数
,使得不存在正整数
满足
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a573743c22f1988094a801651af5611e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1065440ee09654f97b50b5ef6a963ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ef7fb812f8cb384ae86e75fb949ac66.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30de4943aa77db7dc0b92b29b2aa93f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/208400bfcdb85a19359c14f2c66e17c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37e5531913e2f170465d8df01795cd51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f35f7dcce39f3d4dc6b7faf84dc1d0a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86fde3541708c770e48a06c28f9a3434.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbf09398436b9b00458c5d9b245ba287.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c15a081508b98e5c53ebedc4be56c45.png)
(2)求由
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8be646cd52d7f2f1714e7542e75810f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adad9633b73dfbbb3d84b4f15979e99e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37e5531913e2f170465d8df01795cd51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ffb021aa7d5a5c2f0691e337caad624.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b51c3c1cbed76bbb68d50f7df7209a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c45c1e00ea3205d39449f3f9d64ec126.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(3)若实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa7f3694402a60c3b50e39a76d87c90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8a28b156b7cb863056fda24d66fff68.png)
您最近一年使用:0次
8 . 设数列{an}满足:
,n=3,4,…….求证:数列{an}的每一项都是正整数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/decfd4acc3e34538a242a77844e15770.png)
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9 . 设
.求证:
(1)
;
(2)
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dab799bf7d8290d0b33931ac58bf4bae.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04951e8f8d642b4d7fa2e7ad21f1b5f1.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa636f1d62d1656977b079f46832bf81.png)
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