解题方法
1 . 设数列
的前
项和为
,若
.
(Ⅰ)证明
为等比数列并求数列
的通项公式;
(Ⅱ)设
,数列
的前
项和为
,求
;
(Ⅲ)求证:
.
![](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/9ca9605501cceb252348510d860f07c7.png)
(Ⅰ)证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c895d4ce5ce82ef9b311b9369b4de11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(Ⅱ)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b008e35e4367db818d464d31bd2248c.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/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
(Ⅲ)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbfc690ace28306596f1fa5c88fa3c3d.png)
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2020-12-14更新
|
2192次组卷
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8卷引用:【新东方】415
(已下线)【新东方】415浙江省强基联盟2020-2021学年高二上学期期中数学试题(已下线)专题08 数列的通项、求和及综合应用 第一篇 热点、难点突破篇(讲)-2021年高考数学二轮复习讲练测(浙江专用)(已下线)专题4.3 等比数列(B卷提升篇)-2020-2021学年高二数学选择性必修第二册同步单元AB卷(新教材人教A版,浙江专用)(已下线)第4章 等比数列(A卷·夯实基础)-2021-2022学年高二数学同步单元AB卷(苏教版2019选择性必修第一册)【学科网名师堂】(已下线)专题08 数列的通项、求和及综合应用(讲)--第一篇 热点、难点突破篇-《2022年高考数学二轮复习讲练测(浙江专用)》河南省南阳市邓州春雨国文学校2022-2023学年高二下学期3月考试数学试题上海师范大学附属中学闵行分校2023-2024学年高二上学期期中数学试题
2 . 数列
中,
,
(
为常数,
1,2,3,…),且
.
(1)求c的值;
(2)求证:①
;②
;
(3)比较
+
+…+
与![](https://staticzujuan.xkw.com/quesimg/Upload/formula/894ef47a76331c0bf23391e719ad10a6.png)
的大小,并加以证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d4040201a51683b26e158ea278bf0ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e4cc00c283519973f7f8e1274b5c733.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f3cb8d72bb2e281b943b3b430138ef7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5e50b3ac0bec9378d12d82a0b54e83f.png)
(1)求c的值;
(2)求证:①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c610840fda4a40c864e19bc762b385fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/162c0897cdfa67cac5b49becac685e75.png)
(3)比较
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/252de0a549286d1b1721ae96d5832654.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a03fee94205c63211128cbadfa17b810.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b7e761be88728b3db50c2abd4377c12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/894ef47a76331c0bf23391e719ad10a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/090426eb29836bc30c006b3739c08057.png)
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名校
解题方法
3 . 对于数列
,若从第二项起,每一项与它的前一项之差都大于或等于(小于或等于)同一个常数d,则
叫做类等差数列,
叫做类等差数列的首项,d叫做类等差数列的类公差.
(1)若类等差数列
满足
,请类比等差数列的通项公式,写出数列
的通项不等式(不必证明);
(2)若数列
中,
,
.
①判断数列
是否为类等差数列,若是,请证明,若不是,请说明理由;
②记数列
的前n项和为
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7936359df4c926b72b48c6fdae55f12d.png)
(1)若类等差数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc8b1261de54b824c12b6887053416c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14835bf3f00139ccec0694d0924db795.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b0566ce71a91f5939b92eb8d59e8ec5.png)
①判断数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41cf1da18d91f7c98086553d157d1a87.png)
②记数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c806dc9bf2cad0cb20220d23bd252a2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29858a858c8ec1e1c65db718400a4a95.png)
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2022-07-17更新
|
774次组卷
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6卷引用:四川省成都市双流区2021-2022学年高一下学期期末数学试题
四川省成都市双流区2021-2022学年高一下学期期末数学试题(已下线)4.2.3 等差数列的前n项和-2022-2023学年高二数学《基础·重点·难点 》全面题型高分突破(苏教版2019选择性必修第一册)上海市七宝中学2023届高三下学期开学考试数学试题(已下线)4.2.2.1 等差数列的前n项和公式(练习)-2022-2023学年高二数学同步精品课堂(人教A版2019选择性必修第二册)(已下线)4.1 等差数列(第2课时)(十三大题型)(分层练习)-2023-2024学年高二数学同步精品课堂(沪教版2020选择性必修第一册)(已下线)专题03 等差数列(二十三大题型+过关检测专训)(4)
名校
解题方法
4 . 设数列
满足
.
(1)求证:数列
为等比数列;
(2)若数列
满足
,是否存在实数
,使得数列
是单调递增数列?若存在,求出
的取值范围;若不存在,说明理由.
(3)对于大于2的正整数
(其中
),若
、
、
三个数经适当排序后能构成等差数列,求符合条件的数组
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5216eb8b6a878a4a06f4896470ce10ec.png)
(1)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13dc5b2e4b20bfd4841684771a572f10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
(3)对于大于2的正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3531932ad5c05cee5c51c43f7abfef46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8edfabf4c1120b945b6344f60fab63c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff7bd080401c9d37a3bde2d292e5ffe0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87cf58a39b00433d2ffbf34e86ca2f36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbb154d379cea257b00f0df1342d91f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32c7068c999c23b741a3bb15eb0c9e21.png)
您最近一年使用:0次
2021-12-03更新
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1448次组卷
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5卷引用:上海市金山中学2021-2022学年高一下学期期末数学试题
上海市金山中学2021-2022学年高一下学期期末数学试题江苏省苏州市常熟市2021-2022学年高二上学期期中数学试题辽宁省大连育明中学2022-2023学年高三上学期期中考试数学试题辽宁省大连市大连育明高级中学2022-2023学年高二下学期期中数学试题(已下线)第4章 数列 单元综合检测(难点)-2023-2024学年高二数学同步精品课堂(沪教版2020选择性必修第一册)
5 . 已知数列
中,
,
.
(1)证明数列
为等差数列,并求数列
的通项公式;
(2)若
,求数列
的前n项和
;
(3)若存在
,使得
成立,求实数k的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf51f1ab47ee05bb866c7c9fa1c7c3e1.png)
(1)证明数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc0b53ddd01ed8617540f85ce89ce82d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b2aa450fb342a9ea5460c7f1b2d67ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
(3)若存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e145b6046bc80d0ffecc61ac67c87ca1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3805756c0de1bf20101fbe5838650e9d.png)
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名校
6 . 设数列{
}满足![](https://staticzujuan.xkw.com/quesimg/Upload/formula/443e98a6a43e6f72733eb44578ea5411.png)
(1)求{
}的通项公式;
(2)若
求证:数列{
}的前n项和
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/443e98a6a43e6f72733eb44578ea5411.png)
(1)求{
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f3fafe9c68fcd07c5f809726629c3e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72c06a15ed4e4fbdff4d128abb3b01ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10c848b49ceffcd2f60f964068443870.png)
您最近一年使用:0次
7 . 已知数列
的前
项和
满足:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56100cf26e1f761c06b3e5a00bbe1362.png)
(1)求证:数列
是等比数列并写出
的通项公式;
(2)设
如果对任意正整数
,都有
,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.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/56100cf26e1f761c06b3e5a00bbe1362.png)
(1)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f331591a8a32f3e781af90af3a53154.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3dc8aa9fac1a3ac548862699140782f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd624bda9f45309816fc1e6f27e42675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
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2021-01-13更新
|
1224次组卷
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6卷引用:江西省宜春市万载中学2021-2022学年高一下学期第三次月考数学(文)试题
江西省宜春市万载中学2021-2022学年高一下学期第三次月考数学(文)试题江苏省无锡市江阴市成化高级中学2020-2021学年高二上学期第一次月考数学试题人教A版(2019) 选修第二册 突围者 第四章 第三节 课时1 等比数列的概念(已下线)专题05 《数列》中的解答题压轴题(2)-2021-2022学年高二数学同步培优训练系列(苏教版2019选择性必修第一册)福建省连城县第一中学2022-2023学年高二上学期第一次月考数学试题安徽省滁州市定远县育才学校2021-2022学年高二分层班下学期第二次月考数学试题
2011·浙江杭州·二模
8 . 已知数列
、
的各项均为正数,且对任意
,都有
、
、
成等差数列,
、
、
成等比数列,且
.
(1)求证:数列
是等差数列;
(2)求数列
、
的通项公式;
(3)设
,如果对任意
,不等式
恒成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/090426eb29836bc30c006b3739c08057.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/090426eb29836bc30c006b3739c08057.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e95931effbd59c43e8ed1ea09962b84f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e207e0e541808381ccd1c3dbcc7a63a.png)
(1)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b34edecf041aa8544ece5105aa4b8ec.png)
(2)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4fa34d5a86d929757c2bc3db1a51e15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54c76d87cc6647ba4a0d3e402c872ca5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
您最近一年使用:0次
2020-12-14更新
|
978次组卷
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17卷引用:【全国市级联考】浙江省嘉兴市2018年高一下数学期末复习卷三
【全国市级联考】浙江省嘉兴市2018年高一下数学期末复习卷三上海市南洋模范中学2018-2019学年高一下学期期末数学试题陕西省榆林市第一中学2021-2022学年高一下学期期末文科数学试题陕西省榆林市第一中学2021-2022学年高一下学期期末理科数学试题(已下线)2011届浙江省杭州市高三第二次教学质量考试数学理卷(已下线)2012届广东省湛江二中高三2月月考理科数学2015届天津市南开中学高三第四次月考理科数学试卷2016届上海市宝山区高考二模(理科)数学试题2016届上海市长宁、青浦、宝山、嘉定(四区)高考二模(理)数学试题2016届上海市(长宁、宝山、嘉定、青浦)四区高三4月质量调研测试(二模)(理)数学试题2020届上海市高考模拟1数学试题江苏省南通市如皋中学2020届高三创新班下学期第一次高考模拟冲刺数学试题上海市进才中学2021届高三上学期12月月考数学试题上海市青浦高级中学2022届高三下学期4月线上质量检测数学试题上海市青浦高级中学2022届高三4月质检数学试题上海市交通大学附属中学2023届高三下学期卓越测试数学试题(已下线)信息必刷卷04(上海专用)
名校
解题方法
9 . 设数列
的前n项和为
,且满足
,
,数列
满足
,且
.
(1)求数列
和
的通项公式;
(2)设
,数列
的前n项和为
,求证:
;
(3)设数列
满足
(
),若数列
是递增数列,求实数
的取值范围.
![](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/b3d4fe74f5287a2e23d9e8912714f1cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29d5ec9ad92f37e64eccce922ab1b14e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59dd6c97d2ee3e74ba5730f1cbcc1d43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb27cc29c836ab7b82ad4a3acde8a3f5.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b648243c95323ea726ba24a42355cc8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b648243c95323ea726ba24a42355cc8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a36f16deba370e8803480fd670abf44a.png)
(3)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49e9c042584ed68bd73e1901ac594a66.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29d5ec9ad92f37e64eccce922ab1b14e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
您最近一年使用:0次
2020-09-16更新
|
519次组卷
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3卷引用:黑龙江省牡丹江市海林林业局第一中学2019-2020学年高一下学期第一次月考数学试卷
10 . 已知数列
满足
,
.
(Ⅰ)求证:数列
为等差数列,并求数列
的通项公式;
(Ⅱ)若数列
满足
.
①求证:
;
②求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d0ae8af1b4dfc31c317fcbe291d28b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d971b3e74014e2a8eb7e90f4529b42f4.png)
(Ⅰ)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/099a64d86bd0b4602578d910322adc1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(Ⅱ)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a198acf03cdeefbf200ae41fbf577c89.png)
①求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47f8f1b85895c28178f41fd154f76c5d.png)
②求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35ddf0bae58dd1e841cd37847afa8c6f.png)
您最近一年使用:0次
2020-05-26更新
|
1171次组卷
|
2卷引用:安徽省合肥市第六中学2019-2020学年高一下学期期末数学试题