1 . 已知各项均为正数的数列
满足
,且
,
.
(1)证明:数列
是等差数列;
(2)数列
的前项
和为
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7982a20f15d27117b40f6dc6283bdbea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18d8e8f821111de8075e5c3dfb22a5d6.png)
(1)证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c133f850a40f4d23c30fa91a1e7d74a2.png)
(2)数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19bd36c2e9b9b3cd4bc9e65f903a2e43.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/94351ce858fa3f3a09cfadc2d23d7253.png)
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2021-02-21更新
|
126次组卷
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2卷引用:湖南省益阳市桃江县第一中学2020-2021学年高二(研学班)下学期入学考试数学试题
名校
解题方法
2 . 已知数列
满足
,
.
(1)求证数列
为等差数列,并求数列
的通项公式;
(2)设
为数列
的前
项和,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/352d9b76dcf639368fa68cae70149802.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1e6fd3a5e8c59d1fe0813ba38b36989.png)
(1)求证数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41cf1da18d91f7c98086553d157d1a87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7706e0dba93c9f25c28bc8b01de44b70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ed2d714a80009b6cb2c2e62b85fee90.png)
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2020-09-19更新
|
932次组卷
|
2卷引用:重庆市西南大学附属中学2020-2021学年高二下学期第三次月考数学试题
2020·全国·模拟预测
3 . 已知数列
满足
.
(1)求证:数列
是等差数列,并求数列
的通项公式;
(2)设
,
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/066e33f9cc2b11c2ea41ab26cdf84693.png)
(1)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68ffb5a730f63c06263f86e1dcd14e7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/392a8f641017f5df07890012bd80eec6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6ae9b03efb8cf028ba72af8e33f3742.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/183e7c16da08a755ddd79300e2f494d4.png)
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名校
解题方法
4 . 设数列
满足
,
,当
.
(1)计算
,
,猜想
的通项公式,并加以证明.
(2)求证:
.
![](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/2693734765399876e9e93cdb110231c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae90697c66be9e17437eaec2feaf0bd0.png)
(1)计算
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daf464629fa321a6ff7401ab79f07083.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4168dc07f0db5540afc55f886b2ab069.png)
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2020-10-11更新
|
950次组卷
|
3卷引用:安徽省黄山市屯溪第一中学2020-2021学年高二下学期期中理科数学试题
5 . 已知数列
满足
.
(1)证明数列
是等比数列,并求数列
的通项公式;
(2)数列
满足
,
为数列
的前
项和,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56e9503b83ec3fa2939923ae5e4d6902.png)
(1)证明数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8da2e317b095db07efdfa8bea95e3b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a2c71bf821df60553783704f41cd6c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d8dc623a9bac29298adee9a51208790.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9928e46511e601913619a427ded84a3.png)
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2017-11-13更新
|
469次组卷
|
3卷引用:甘肃省兰州市永登县第一中学2020-2021学年高三上学期期末数学(文)试题
6 . 已知函数
(
是自然对数的底数,
).
(I)证明:对
,不等式
恒成立;
(II)数列
的前
项和为
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/331d70266454df40256268b19b055a88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168b3e4b1d6f04226fa2687a72a268b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d218992d1942266d7208e476d0c4100.png)
(I)证明:对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c780149aef1bd77162e85f7f8906a6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04a6a4357fbdb4015810df156e1ed559.png)
(II)数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfbae2b0b08f55a23cea77f388381276.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/21a7305b8d7a0930e10b454e3a48bbd5.png)
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名校
7 . 在单调递增数列
中,
,且
成等差数列,
成等比数列,
.
(1)①求证:数列
为等差数列;
②求数列
通项公式;
(2)设数列
的前
项和为
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5786daa387797fe28543eb25cdcf0193.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48aa8f272b068a13e9a61912ed5697cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee635f30f8c1ab7cc90ca44ea5071f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cd2bfef3925d6f9f46b96b301c58223.png)
(1)①求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4fffabc2dfb59ac198c06dbcadfa75c.png)
②求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
(2)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a44cfbb86a4eb76261c00ddc6bff181.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/9fea6ba08b4985e51979378af23595d5.png)
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2016-12-04更新
|
970次组卷
|
4卷引用:黄金卷13-【赢在高考·黄金20卷】备战2021年高考数学(文)全真模拟卷(新课标Ⅱ卷)
(已下线)黄金卷13-【赢在高考·黄金20卷】备战2021年高考数学(文)全真模拟卷(新课标Ⅱ卷)2017届河北衡水中学高三上学期第二次调研数学(理)试卷2016-2017学年湖北省孝感市七校教学联盟高一下学期期中考试数学(理)试卷河北省保定市定州中学2021届高三上学期期中数学试题
名校
解题方法
8 . 公比为
的等比数列
的前
项和
.
(1)求
与
的值;
(2)若
,记数列
的前
项和为
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.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/e6b981758450e9dcee6cfbe6c67c61f8.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0f6000421c5370e4b89f23be199f388.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/05f5b1a6c081ca11ee5c4723525a43ce.png)
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2024-01-16更新
|
1318次组卷
|
3卷引用:湖南省株洲市第二中学2022届高三上学期期中数学试题
名校
解题方法
9 . 已知等比数列
的公比
,且
,
,
是公差为
的等差数列
的前3项.
(1)求
的最小值;
(2)在
取最小值的条件下,设
,数列
的前
项和为
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eda6dc559d07bc22c9a0ed1e3a6d01d2.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/fc17ca3ab612ea9cf6cfa1eea53cb1eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/450bfba8c76f5957e945026cbd235298.png)
(2)在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/450bfba8c76f5957e945026cbd235298.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7f672010fe85e005afba869d1c50e27.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.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/2e8aef2efae0e76650c8463d80f69a14.png)
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2024-01-13更新
|
311次组卷
|
2卷引用:湖南省长沙市宁乡市第一高级中学2021届高三第三次模拟考试数学试卷
名校
解题方法
10 . 已知正项数列
的前
项和为
,
,且
.
(1)求数列
的通项公式;
(2)若数列
满足
,设数列
的前
项和
,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/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6b0e5d689559e2c2c3a44017cd1462d.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/809340ac7c2e74a19828f91ab4d2373e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d589bd43d1bc63e7e77120747e4cfa4.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/e672c3d231081d12e44a4211e5ac60bf.png)
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