名校
解题方法
1 . 已知数列
的前n项和为
.若对每一个
,有且仅有一个
,使得
,则称
为“X数列”.记
,
,称数列
为
的“余项数列”.
(1)若
的前四项依次为0,1,
,1,试判断
是否为“X数列”,并说明理由;
(2)若
,证明
为“X数列”,并求它的“余项数列”的通项公式;
(3)已知正项数列
为“X数列”,且
的“余项数列”为等差数列,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f6b18b109a656b62fb173680ae99ca7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a42dd37c118e64c46c7fc37e21081745.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/450706c32e58d9e6ad2f14aabf9e81ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.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/acbc6a613224461ade69362d46550474.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d255ea8e125b603d6b640bdf4a804922.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)已知正项数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/771ca8c38c8a1646c83481a1d2bcfdfa.png)
您最近一年使用:0次
2024-05-07更新
|
1429次组卷
|
3卷引用:黑龙江省牡丹江市第三高级中学2024届高三下学期第四次模拟数学试卷
2 . 差分法的定义:若数列
的前
项和为
,且
,则
时,
.例如:已知数列
的通项公式是
,前
项和为
,因为
,所以
.
(1)若数列
的通项公式是
,求
的前
项和
;
(2)若
,且数列
的前
项和分别为
,证明:
.
![](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/314fa1f4da470780673cc7246974180c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9361afc7cc02253140585eedc39a695d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/677e46ecd051c92489c0d1d458932f37.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/e3bd2e55bb083a90ecba8cc98fac9536.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/237ce153a42d4e2378d5435051734cb3.png)
(1)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bd845d1bfac72200926447db04563fc.png)
![](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)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77af844c4444e536adae9bc0b1cff614.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c04f062dc12653209868713f2142fe06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee1c51f15c934050099b460b19a04f4b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/038e3af7c9f2fb642b9209415662aeff.png)
您最近一年使用:0次
2024-05-30更新
|
159次组卷
|
2卷引用:黑龙江省伊春市铁力市第一中学校2023-2024学年高二下学期期中考试数学试卷
名校
解题方法
3 . 已知点
为线段
上的点,点
为
所在平面内任意一点,
,
,
,
,设
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/20/dbf32769-0600-418a-becc-15f28f50b0cd.png?resizew=193)
(1)求证:
,并求出
的值;
(2)若
,求
的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39acab3cfb59bfc9591371721ab01d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca036d049f5205cf04cb1b9c5cd03f97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85feff0ce62d9a89db8fc47b5952d5da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/262e062dcdd2039084a356862b123e9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc48a3935a66d6fcdc0aaa3e6a331c38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49128b2e5ecfd4aea19453f0bd52b3fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75fa3a75aacb99a46f46c229ec094a38.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/20/dbf32769-0600-418a-becc-15f28f50b0cd.png?resizew=193)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34377f5318447ea6e2e7d5f4b126d5bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ccc2f02416db8211128e18af2d13ecf.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12133283c743f2a43f0416beb30c1975.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2205cffebf8c4d5f81d15ed7b85c8936.png)
您最近一年使用:0次
2023-05-18更新
|
527次组卷
|
2卷引用:黑龙江省哈尔滨市第三中学校2022-2023学年高一下学期期中数学试题
名校
4 . 射影几何学中,中心投影是指光从一点向四周散射而形成的投影,如图,
为透视中心,平面内四个点
经过中心投影之后的投影点分别为
.对于四个有序点
,定义比值
叫做这四个有序点的交比,记作
.
;
(2)已知
,点
为线段
的中点,
,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42c2d86d8daea5e652d99fe1c6bc3f9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c82a10b4f0c9323d726804c89dd9548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c82a10b4f0c9323d726804c89dd9548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d33747c77ff8ec31b1d8787a2a99748.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20fc6388f7dd9e393808bfcfb41b499e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19d4c674a3fe91bd4bffd3dcd9ea58f1.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29998510e4ecded4acfc9e981da9110f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/080ead0dc6f5e5881fd26b1a07f37024.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b32f2d4d1d2c16c54b2caef17840bfcb.png)
您最近一年使用:0次
2023-07-11更新
|
979次组卷
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10卷引用:黑龙江省鹤岗市工农区鹤岗市第一中学2023-2024学年高三上学期开学数学试题
黑龙江省鹤岗市工农区鹤岗市第一中学2023-2024学年高三上学期开学数学试题山东省济南市2022-2023学年高一下学期期末数学试题(已下线)模块四 专题5 暑期结束综合检测5(能力卷)(已下线)专题22 新高考新题型第19题新定义压轴解答题归纳(9大题型)(练习)(已下线)第11章 解三角形 单元综合检测(难点)--《重难点题型·高分突破》(苏教版2019必修第二册)吉林省长春市东北师范大学附属中学2023-2024学年高一下学期5月期中考试数学试题(已下线)上海市高一下学期期末真题必刷04-期末考点大串讲(沪教版2020必修二)(已下线)专题01 平面向量及其应用(2)-期末真题分类汇编(新高考专用)【人教A版(2019)】专题09解三角形(第三部分)-高一下学期名校期末好题汇编(已下线)重组2 高一期末真题重组卷(山东卷)B提升卷
5 . 余弦定理是揭示三角形边角关系的重要定理,也是在勾股定理的基础上,增加了角度要素而成.而对三角形的边赋予方向,这些边就成了向量,向量与三角形的知识有着高度的结合.已知
,
,
分别为
内角
,
,
的对边:
(1)请用向量方法证明余弦定理
;
(2)若
,其中
为
边上的中线,求
的长度.
![](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/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.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/c5db41a1f31d6baee7c69990811edb9f.png)
(1)请用向量方法证明余弦定理
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34369422d71dd95c61cdd1b8245d7b6c.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48897a577999a24e15e8645e7b23e592.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
您最近一年使用:0次
2023-06-11更新
|
629次组卷
|
4卷引用:黑龙江省大庆市大庆铁人中学2022-2023学年高一下学期期中数学试题
名校
解题方法
6 . 已知等差数列
的前
项和为
,首项为
,
.数列
是等比数列,公比
小于0,且
,
,数列
的前
项和为
,
(1)记点
,证明:
在直线
上;
(2)对任意奇数
恒成立,对任意偶数
恒成立,求
的最小值.
![](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/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3de34c34529c9725beacee13db7b4e42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/371473cfe69603fea1895c793219e3e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e18c9b373494f7eb0128748120c1fdaa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ec0699dc1a6f668a50c3017116751e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(1)记点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5fe6bf7695ea9b0a129d9f01953360e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5a2d3cd8e283ae9d04bee5ab2e0895b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fa0667ad7e7060032787706750f409e.png)
(2)对任意奇数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7db29df7dc5fdc524c6de316ac2a04cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bc489c026bcc13f539e1b22467ee8b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5eeaa30850a2dd1abcb46208ac8b6a2.png)
您最近一年使用:0次
7 . 已知数列
的前
项和为
,
,给出以下三个命题:
①
;②
是等差数列;③![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28babe6d68eb0646f9e97bd9e01c3c86.png)
(1)从三个命题中选取两个作为条件,另外一个作为结论,并进行证明;
(2)利用(1)中的条件,证明数列
的前
项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99aed0821e3241d550edfdc7c329cf50.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)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6e35ba37b0cc1d390e05391554e9660.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99aed0821e3241d550edfdc7c329cf50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28babe6d68eb0646f9e97bd9e01c3c86.png)
(1)从三个命题中选取两个作为条件,另外一个作为结论,并进行证明;
(2)利用(1)中的条件,证明数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fa2c40028103fd23931f8772b0cefa9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c215db1d8f69757118ad405b78035628.png)
您最近一年使用:0次
2022-02-22更新
|
679次组卷
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3卷引用:黑龙江省双鸭山市第一中学2021-2022学年高三上学期期末考试数学(理)试题