2004高三·福建·竞赛
1 . 无穷数列
中,对每个奇数
成等比数列,二队每个偶数
成等差数列.已知
.
(1)求数列的通项公式,实数
满足怎样的充要条件时,存在这样的无穷数列?
(2)求
的调和平均值,即
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e2eff04a2f9a8d85847cf41ff71e627.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50f27f993b10cc5f4e2e2ca356eccfa2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50f27f993b10cc5f4e2e2ca356eccfa2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dc1fc8aa9f6d6cbf67608dd916507c4.png)
(1)求数列的通项公式,实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/663a61ad241d5d874c9a9362f0ee917c.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00bc48e2c83dc67de5438a1fd36154b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/170c9f38dbb277d23c476a979a3f7bd9.png)
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2 . (1)给定正整数
,集合
,是否存在一一映射
满足条件:对一切
,都有
?
(2)
为全体正整数的集合,是否存在一一映射
满足条件:对一切
,都有
?证明你的结论.
注:映射
称为一一映射,如果对任意
,有且只有一个
,使得
题中“丨”为整除符号.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76238e269c78a50786c7d802f38d9244.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0460251d1b3dbbb6883e6e2c7e71b0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc8d74bb3cf91246b56bbb5b584b0c97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d61bd1212fab288ce1511ca367d231f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d557756cc1cf687c9e50908425272cc5.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81995b5247fc02e27b6997dc7c0219bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16b95693197b1a64474d46a57c991fa1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4cfe7dea265c4dd8ab396a774e91eb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d557756cc1cf687c9e50908425272cc5.png)
注:映射
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d869ac78a02498dd95f0d67990c97dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d4ad551801ac85b50651304411dd59c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7c3e3323c13b1ddc0e0869a6a6e6056.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea97b95974dc37718ef97eca2ace1dd1.png)
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