1 . 已知数列
的前
项和为
,
且
.
(1)证明:
,并求
的通项公式;
(2)构造数列
求证:无论给定多么大的正整数
,都必定存在一个
,使
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/675668080c75d532a01242ab465b411c.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fb8d673ef472670f69449a512cc4449.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)构造数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d90b7178ae1c099f5cf8c8c9273cc04f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0475bd436d7637c7ad6a18e13ad66d64.png)
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2 . 证明:
(
且
).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89acdb8232fe9ed19439fdc15c8f4070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4591f545ee3ce704391e55da4ad940d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/856b137a34d2d5b20671b7a3c7a29606.png)
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3 . 设数列
满足
.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df2577a0871cb3b018eb5e2840da1d7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48d5780c707bbf8fc91265df5f2093cb.png)
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4 . 已知n个非负实数
和为1.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fe1c31a81f198c443e71b83ca662939.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/493f509c3370b2b26d46e2e46d038134.png)
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5 . 已知数列
满足:
,且对于任意正整数
,均有
.
求证:(1)
;
(2)数列
为单调数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3270b2da0e77dbdd6d3c6ad86f28c206.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/273e9f34be39e9ae61043721d9c14577.png)
求证:(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5749e4e6ecbead28a9027f2f687649cc.png)
(2)数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/467dbb09f25fc04c974f2be5e66405d8.png)
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6 . 已知数列
满足
.
(1)求证:
.
(2)是否存在实数
,使得
,若存在求出
的值;若不存在.请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dffe4a726c5d5c2c1b51255dd6e9015.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d41ebd083ba2fdca7e830d62e3d5181b.png)
(2)是否存在实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0920804135bf9f7a222df640a6e874ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
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7 . 已知
.证明:当
时,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d1a4f6aed102bf6a4ac8b0f7dc63228.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/152b9780c93f0157f48c59a946354d37.png)
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8 . 设
是整数.对每个正整数
,令
为
在
进制表示下的非零数字的个数.证明:对于任意给定的正整数
和
,存在正整数
使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/731bdc8d2686a05f12a2ba8a7e3b01be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d1748a0e5fba63eac995e4388bb537.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5809edaae5fd443ff3126d8ad0eaa1b5.png)
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9 . 设
和
为两组复数,满足:
.求证:存在数组
(其中
),使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40b2f1f2faeb04505752c4c08b5d4a24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5b023415e27a14faac81fd5237fa1a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38af3182c396bef07007d4b04c92d32c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f094ba98f6c02216c714bd9ff85bd59f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/057d57358dd4fd992df26157efa5bd9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83c72aec1d89b300f5a069080cdd3e1a.png)
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