2024·全国·模拟预测
1 . 设
,
,…,
(
),
,
,…,
(
)为两组正实数,
,
,…,
是
,
,…,
的任一排列,我们称
为这两组正实数的乱序和,
为这两组正实数的反序和,
为这两组正实数的顺序和.根据排序原理有
,即反序和≤乱序和≤顺序和.则下列说法正确的是( )
![](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/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdd14381552c8f3a896675effe1f4f4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b715e7842b95f654f16056a7c7f2abe9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13423c094861baf4b759b7f3d8c3c226.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/473bae0bc5acfd6a486c1433c58fb369.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7936359df4c926b72b48c6fdae55f12d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b76f79be89b8c6227b68eded6b675546.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c59e7c7a84a4bdb959e95536d0404ceb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b715e7842b95f654f16056a7c7f2abe9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13423c094861baf4b759b7f3d8c3c226.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a71581a9193c69e5d4f644e6100ffd34.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6214883c5996a384348d3052f05dd2b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6afad2abfe7cfccd0d634c1d8951121b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a357cbf7931e33754ae34bca26b98ac3.png)
A.数组![]() ![]() |
B.若![]() ![]() ![]() ![]() ![]() |
C.设正实数![]() ![]() ![]() ![]() ![]() ![]() ![]() |
D.已知正实数![]() ![]() ![]() ![]() ![]() |
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解题方法
2 . 伯努利不等式又称贝努力不等式,由著名数学家伯努利发现并提出. 伯努利不等式在证明数列极限、函数的单调性以及在其他不等式的证明等方面都有着极其广泛的应用. 伯努利不等式的一种常见形式为:
当,
时,
,当且仅当
或
时取等号.
(1)假设某地区现有人口100万,且人口的年平均增长率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ba01d85cd57bded85cf3302538084bd.png)
(2)数学上常用
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cca374b4e6d3ebc183c5b21d4ea7220.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)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cc69c193ab6d75fcb9152f513a681f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
(ⅰ)证明:;
(ⅱ)已知直线与函数
的图象在坐标原点处相切,数列
满足:
,
,证明:
.
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2023高三·全国·专题练习
3 . 已知m,n为正整数.
(1)用数学归纳法证明:当
时,
;
(2)对于
,已知
,求证:
;
(3)求出满足等式
的所有正整数n.
(1)用数学归纳法证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adc3e5be1796493161a4df7e28a6f6b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4b6e1ca4dd76952e3544b14978bd359.png)
(2)对于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/831608f09609c37f757f5bfcd01253f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/186c794ebbde3237056af29cb97778f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91cbed0668c7d57785e9d95b5775e41f.png)
(3)求出满足等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91192b519c2eb79c20d0205bc22016a2.png)
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2023高三·全国·专题练习
解题方法
4 . 已知
,且
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a521891098b625f372ff648d110afe1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/574b3c0cdd4e1c6f3c77d43dc7e0603f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ec4399896254d7506dc73c51653e1a.png)
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2023高三·全国·专题练习
5 . 已知
,
,
,且
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cec12441802f71e803efaf2c62ee588.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/141e9389230c8bcdaa24c43893b3d192.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/624573cd4212cbab34626d77b15042a7.png)
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2023高三·全国·专题练习
6 . 若a,b,c为任意的正数,则有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3bba19a7559d184d6e0dec74adceb54.png)
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2023高三·全国·专题练习
解题方法
7 . 若
,则有
(1)
;
(2)
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edcff1dd54bc8cf631e2f804b6e10282.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a601d082dc0998fc5eede6542c4af590.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbda5cadd4dd2d7d94b4b289fb5805ab.png)
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2023高三·全国·专题练习
解题方法
8 . 设
,且
,求证:
. 推广:设
,且
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d05bf789a20dbfced92873a2198dfbc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/751e274e9107d780c39ba9c49d6daefb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e17c70c17e5ea42d299412c253c2cbc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24843cd1bf2ef644d5fa84ca35b193d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b16ad49f62d7362441e3b92efe7f87d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55601eaaec7911c9efa12f3004ece63a.png)
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2023高三·全国·专题练习
9 . 设
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37698188641bac67421d9b426cd3c3ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52ef977076b58c91d1d8e7b9d7fcbd05.png)
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2023高三·全国·专题练习
10 . 证明不等式
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4da4d70e203ac268b644aee388ecd63.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
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