2023高三·全国·专题练习
1 . 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
2 . 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
3 . 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
4 . 设
,
且
,求证:
等号成立当且仅当
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ebdf5e12eff022da3423b54d6fc16a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59b446bc370b54239bbf1c5407464894.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57db73d5e95d0dda392d5af3a5cecaa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79aca8d6f8787ae1bab1a08534a1de86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2697ef67790838c84cc238a0334c5d47.png)
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2023高三·全国·专题练习
5 . 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
6 . 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
7 . 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
8 . 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
9 . 证明数列
,
单调递增,而数列
单调递减.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba9fccf592b578a91aa576e06d61d012.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5416ab259583616b6d6d2383735ef07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4a27ecebfd60eb193e95a60ba96babf.png)
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