2023·全国·模拟预测
1 . 在数列
中,
,
.
(1)证明:数列
是等比数列;
(2)令
,数列
的前n项和为
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a4a90f7b0e4b2a39bea76fc2efc58e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f756b4a1896a2677a77aa8cfa8312137.png)
(1)证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c895d4ce5ce82ef9b311b9369b4de11.png)
(2)令
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e16ff08c8a2a1011826b41e3a12eaea2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e30136113176ba7fe660e998d0873157.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bee5ea120d7c0ca997845c9cc77772fc.png)
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6卷引用:2023年普通高等学校招生全国统一考试数学预测卷(九)
(已下线)2023年普通高等学校招生全国统一考试数学预测卷(九)辽宁省铁岭市清河高级中学2022-2023学年高二下学期3月月考数学试题(已下线)专题15 数列求和-2辽宁省辽东十一所重点高中联合教研体2024届高三第一次摸底考试数学试题山西省大同市第一中学校2024届高三上学期10月月考数学试题山东省德州市临邑第一中学2023-2024学年高三10月月考数学试题
解题方法
2 . 已知各项均为正数的数列
、
满足
,
,且
,
,
成等差数列,
,
,
成等比数列.
(1)证明:数列
为等差数列;
(2)记
,且数列
的前
项和为
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fce83115a50f99e08e9a2db7267aeed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86fc336b4a83bf6d66c4afcc431597f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/385275d29d8c8a7841eaeaa3dfab2cdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e95931effbd59c43e8ed1ea09962b84f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e95931effbd59c43e8ed1ea09962b84f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/090426eb29836bc30c006b3739c08057.png)
(1)证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac3c908fff3de3f31eacff9e2ada4dc2.png)
(2)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01a7e265a92cb2d50eb4628be69668a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38ef4c4439b36c2847b0056a116d56d4.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/105de1b20942840a12712c6795a05e1b.png)
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2022-07-29更新
|
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3卷引用:安徽省黄山市2021-2022学年高二下学期期末数学试题
安徽省黄山市2021-2022学年高二下学期期末数学试题浙江省金华十校2022-2023学年高二上学期期末联考模拟数学试题2(已下线)第四章 数列章末检测卷(二)-【帮课堂】2022-2023学年高二数学同步精品讲义(人教A版2019选择性必修第二册)
3 . 已知数列
满足
记
.
(1)证明:数列
为等比数列,并求出数列
的通项公式;
(2)求数列
的前
项和
.
(3)设
,记数列
的前
项和为
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ead92b9d9d976b90729da850bee2867.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4188680e5320653753ad0340439cb77.png)
(1)证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2d51f9147b8265c0276c1f2c2659197.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b9a0d7150fb24be3e28ef7f0e18be93.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2d27e85083be03c10537535ec96e424.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/c215db1d8f69757118ad405b78035628.png)
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名校
解题方法
4 . 已知数列
中,
,
,
.设
.
(1)证明:数列
是等比数列;
(2)设数列
的前
项的和为
,求
.
(3)设
,设数列
的前
项和
,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/9c9b6e51986fe5d7a7265e0e93adcb4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b3fbc9f52077c96e41e952d3cae583c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5653b60d16ec4e653518f0562680250.png)
(1)证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)设数列
![](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/08eb71ecf8d733b6932f4680874dbbf3.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68e3063fcab5b92188618b4500cb1be4.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/c02e80983b88cdf6b540502816c87d13.png)
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5 . 记
是公差不为0的等差数列
的前
项和,已知
,
,数列
满足
,且
.
(1)求
的通项公式,并证明数列
是等比数列;
(2)若数列
满足
,求
的前
项和的最大值、最小值.
(3)求证:对于任意正整数
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.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/6b106f3aed5e2f23e10c1605045dccbc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/360d929d12ccfdf847e487cf8eeabf38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2669b03c9edf3947bd588e5bb0d800d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca9b0e5214575fdbfbe00302189656f7.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/907fce0e59f19c1dfcad75aceac9572b.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7572ce0d3130c83d0025e1854d63a548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(3)求证:对于任意正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8dd01dc4ac5ae74f09dddd2882bf3b24.png)
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|
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5卷引用:天津市南开中学2022-2023学年高三上学期第二次月考数学试题
天津市南开中学2022-2023学年高三上学期第二次月考数学试题(已下线)专题05 数列放缩(精讲精练)-1天津市微山路中学2022-2023学年高三上学期期末数学试题(已下线)专题6-3 数列求和-1天津市南开中学2023届高三上学期期中数学试题
名校
解题方法
6 . 已知数列
的前n项和
满足
.
(1)证明:数列
是等比数列;
(2)设数列
的前n项和为
,求证:
.
![](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/d3cab1c977bfc938b7b865c16312aacf.png)
(1)证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51e18eb693f55edd2b9f26d3a7010d25.png)
(2)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/203a63ef29b384faa8ee3b7ae870ba2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1056f02e4a7e9b8fd479519eec2d9b3.png)
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2022-07-07更新
|
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6卷引用:山西省大同市2023届高三上学期第一次学情调研数学试题
山西省大同市2023届高三上学期第一次学情调研数学试题(已下线)专题27 数列求和-2(已下线)第7讲 数列求和9种常见题型总结 (2)1.3.2 等比数列与指数函数(同步练习提高版)(已下线)第四节 数列求和 A素养养成卷四川省绵阳南山中学实验学校补习版2023届高三一诊模拟考试理科数学试题
7 . 已知数列{
}的首项
=2,
(n≥2,
),
,
.
(1)证明:{
+1}为等比数列;
(2)设数列{
}的前n项和
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0016094d4c3ee9283e60749616ff6201.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e145b6046bc80d0ffecc61ac67c87ca1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cb23f1b6940b40353ca3d6397d5c21e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1198d9a8c802928d39bb31c6c5db6110.png)
(1)证明:{
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
(2)设数列{
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c59e7c7a84a4bdb959e95536d0404ceb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bbb0a0bb77044ff507eb050821f1f16.png)
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4卷引用:广东省茂名市五校联盟2021-2022学年高二上学期期末联考数学试题
名校
解题方法
8 . 已知数列
满足:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ba2c8c4e2656c84dba72154aa2b980f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/132e9579e58d8d5225e2340e1f43adf1.png)
(1)求
、
、
;
(2)将数列
中下标为奇数的项依次取出,构成新数列![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ad351773d8117faa128041a877bf2db.png)
,
①证明:
是等差数列;
②设数列
的前m项和为
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ba2c8c4e2656c84dba72154aa2b980f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/132e9579e58d8d5225e2340e1f43adf1.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f65fc200f10b97588a0c9896277c9c64.png)
(2)将数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ad351773d8117faa128041a877bf2db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74d0aea7b7bcbd8bf1ef02c406f601ec.png)
①证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be998aceb5c2e14b797271f1cee536d9.png)
②设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/964ae4bf0271ad52323c1135866b3817.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1752474698cd5466dd180df0a00ba9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36304574f1d3bb7e27e4289263abd245.png)
您最近一年使用:0次
2022-06-15更新
|
1434次组卷
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3卷引用:江苏省无锡市江阴市2022届高三下学期最后一卷数学试题
9 . 已知数列
中
,
,
.
(1)证明数列
是等比数列,并求
的通项公式;
(2)记
,
是数列
的前n项和,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1bae03ee4ac75dacfb026290e4207dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/705c826f3b3e084d2fdfcdbac7d18ac3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e145b6046bc80d0ffecc61ac67c87ca1.png)
(1)证明数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d17d72d1d20d385920c3d9da6bed8bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0530a06e7167d07eaf988f4c29f65f2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4358c212e1f7ec327d2b738da67f0135.png)
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2022-04-08更新
|
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3卷引用:辽宁省名校联盟2021-2022学年高二下学期4月联合考试数学试题