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解题方法
1 . (1)
的三个内角
成等差数列,
的对边分别为
.求证:
.
(2)已知:
为互不相等的实数,且
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/043714f337a44c343813c4e34f699211.png)
(2)已知:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a14c388e1e2e5a2ff1ccf6caffbee0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/baa4b450e9269a7ef67582e7359f0125.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27b2d4c175ae8fadf2da3078ec2904d4.png)
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2 . 证明下列各题:
(1)求证:
;
(2)用综合法或分析法证明:若
,则
.
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/add637eef4cd8802b4eb211aa4f6e572.png)
(2)用综合法或分析法证明:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf04fe8895c10624636a815d3d752975.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0da537e5284dc9786845fca39a9ca913.png)
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3 . 把正整数1,2,3,…,n按任意顺序排成一行,得到数列
,称数列
为1,2,3,…,n的生成数列.
(1)若
是1,2,3,…,8的生成数列,记
,数列
所有项的和为S,求S所有可能取值的和;
(2)若
是1,2,3,…,10的生成数列,记
,若数列
中的最小项为T.
①证明:
;
②求T的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9df0d519bd26388e2ab1934625d89bd6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d702451d2c4a01591c0cec57f396faf4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
①证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8e0d245d25d34ce73a7d7d8c2587cd6.png)
②求T的最大值.
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4 . 已知集合
,对于
,
,定义
与
之间的距离为
.
(1)已知
,写出所有的
,使得
;
(2)已知
,若
,并且
,求
的最大值;
(3)设集合
中有
个元素,若
中任意两个元素间的距离的最小值为
,求证
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0a1ddf7f51f115fa436aadb2de92b3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1e6a5b4c75edf2ee68f935b2e3fe12d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec1a7c3713945abc4eca8485945abf32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ab4a9bfa50054c808dd8190305d0abd.png)
(1)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9e25cf1d30f458593f08325f85f6e0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42c72cbaca91d8e578762c4f0b6750a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4aaf68eac26dd95e0eb726913dfc17b.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/003fbb029cdb6d5d7f93e29dca371f7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e77ba8c90d21237670483bbcd8ac63b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c332b53365518c5cf5361bf3cacd47b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce4a2681390214200443ae07c01a4abe.png)
(3)设集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11feb79d33ca429733a82c5d88393ba1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1071ac8657ef1c4e1ea7e0530196298d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9d528f49ab8d7fd971c5bb8fa7f24d4.png)
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5 . 在
中,角
的对边分别是
,且
,求证:角
为锐角.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38335830b93ac4d99c28a8e209eecb3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7f5573b30734d65648f61c0a94c98de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/017812dfd2c0a8dab627a42a5bd8c06d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
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6 . (1)用文字语言和符号语言叙述异面直线判定定理:
文字语言:过______一点和______一点的直线,和此平面上______的任何一条直线是异面直线;
符号语言:若______,则直线
与直线
异面.
(2)用反证法证明异面直线判定定理.
文字语言:过______一点和______一点的直线,和此平面上______的任何一条直线是异面直线;
符号语言:若______,则直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)用反证法证明异面直线判定定理.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/17/0b755a73-3352-4334-8e98-4098033dc32d.png?resizew=137)
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解题方法
7 . 已知数列
的首项
,且满足![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6931285e9564e0edecab772a79db523.png)
(1)求证:数列
为等比数列;
(2)证明:数列
中的任意三项均不能构成等差数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc6545b8eca1c4223ed701a199a85683.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6931285e9564e0edecab772a79db523.png)
(1)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/213e22890204937a5dded4436369390f.png)
(2)证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/213e22890204937a5dded4436369390f.png)
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8 . 用反证法证明时,否定结论“至多有一个解”的说法中,正确的是( )
A.没有解 | B.有一个解 |
C.至少有两个解 | D.至少有一个解 |
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9 . 给出一个命题
:若
,且
,则
中至少有一个小于零,在用反证法证明
时,应该假设( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97fab2940de09665355dd142a9802490.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a08f4e12d723ec259c98b44c5aa1d4a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d10449bc77d692a7270e0f20a68cdf2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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10 . 已知
是无穷数列,
,
,且对于
中任意两项
,
,在
中都存在一项
,使得
.
(1)若
,
,求
;
(2)若
,求证:数列
中有无穷多项为0;
(3)若
,求数列
的通项公式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7fab51121848ce166035ceab6f4e00b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a55ef34345210312db273ab4981c40f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50a272adba0f1120109824440f0e252c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0616dca5cf0229b9f801365cc2bcfff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba50a82a53f0e597c096ccf5746f1b9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a53abaaac2e62f510d996e6db22aefe7.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e65397f11ea8af736f38debadf420c4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23725094c363fd158166a8698971694c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/657435e1fda84118e7f63c97505c8b75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6a46e678bf9d2df5ad4c782b3dc22f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
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