解题方法
1 . 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
解题方法
2 . 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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