1 . 南宋数学家杨辉所著的《详解九章算法·商功》中描述了如图所示的形状,后人称为“三角垛”.三角垛的最上层(即第一层)有1个球,第二层有3个球,第三层有6个球,…,从第二层开始,每层球数与上一层球数之差依次构成等差数列.现有60个篮球,把它们堆放成一个三角垛,那么剩余篮球的个数最少为______ .
![](https://img.xkw.com/dksih/QBM/editorImg/2023/7/6/3d9e2654-60ef-46a8-bfe8-a254b95422d8.png?resizew=116)
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2023-07-06更新
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449次组卷
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3卷引用:上海市延安中学2023-2024学年高二上学期期中数学试题
名校
2 . 已知各项均为正数的数列
的前
项和为
且满足:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2324668a24c3a24b80352ad7ed4cf74.png)
(1)求数列
的通项公式;
(2)设
求
的值;
(3)是否存在大于2的正整数
使得
?若存在,求出所有符合条件的
若不存在,请说明理由.
![](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/117464f527849ab995858aaa20f4175b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2324668a24c3a24b80352ad7ed4cf74.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5091db733b3fd827a8a988060ad4496d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a83259ee12078b65f71a29c05ecc8452.png)
(3)是否存在大于2的正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0726e5ad5e615b083d8d5ca660c31e96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7f7610c827f0fc5017a8c8b6f188242.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4434b36dfbfed1d904b6b1f11e8683a0.png)
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2019-12-07更新
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245次组卷
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2卷引用:上海市延安中学2018-2019学年高三上学期期中数学试题
3 . 已知数列
、
满足:
,
,
,
.
(1)求
,
,
,
;
(2)求证:数列
是等差数列,并求
的通项公式;
(3)设
,若不等式
对任意
恒成立,求实数
的取值范围.
![](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/2438f2272d7b7ab51dbbe587025a553d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4b049addc11b7c1cc622b745ebbd367.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f281aaf2d33d1d7b582f2d84724f5d31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/637ab2b7011537645031dd2191226673.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b715e7842b95f654f16056a7c7f2abe9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13423c094861baf4b759b7f3d8c3c226.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57483e04fd1840c87ac5325157149877.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a548938d87c80ac47910607d3857007f.png)
(2)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de7723c7775e9e3a24e90292d95ffdac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fce83115a50f99e08e9a2db7267aeed.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f347f0643c8013d25ca2d88e87a44a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3458e228949da4c92f5b8cd0173ba7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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2020-02-02更新
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466次组卷
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2卷引用: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)(文)数学试题
4 . 已知数列
的前
项和为
,且
,
(
),若
,
![](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/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cbb0f2883ddd23be9435fda33e9c611.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/508a3cfbc13c723c24918b7f416442e7.png)
则数列的前
项和
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2018-01-20更新
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1632次组卷
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6卷引用:上海市长宁、嘉定区2018届高三第一次质量调研(一模)数学试题
上海市长宁、嘉定区2018届高三第一次质量调研(一模)数学试题(已下线)《2018届优生-百日闯关系列》数学专题 二 第五关 以数列求和或者通项公式为背景的填空题(已下线)《2018届优生-百日闯关系列》数学专题四 专题四第六关上海市上海师范大学附属中学2018-2019学年高三下学期质量检测数学试题2019届新疆乌鲁木齐地区高三第三次质量检测数学(理)试题2018届上海市嘉定、长宁区高三一模数学试题
5 . 已知数列
中,
,
,
的前
项和为
,且满足
(
).
(1)试求数列
的通项公式;
(2)令
,
是数列
的前
项和,证明:
;
(3)证明:对任意给定的
,均存在
,使得当
时,(2)中的
恒成立.
![](https://img.xkw.com/dksih/QBM/2015/5/13/1572106333437952/1572106339598336/STEM/da2e6d541df64409b1532a92c738d498.png?resizew=31)
![](https://img.xkw.com/dksih/QBM/2015/5/13/1572106333437952/1572106339598336/STEM/6572a248423740398d63119f72eea88f.png?resizew=41)
![](https://img.xkw.com/dksih/QBM/2015/5/13/1572106333437952/1572106339598336/STEM/38a40195785b4f45bbde9e84124adb19.png?resizew=43)
![](https://img.xkw.com/dksih/QBM/2015/5/13/1572106333437952/1572106339598336/STEM/da2e6d541df64409b1532a92c738d498.png?resizew=31)
![](https://img.xkw.com/dksih/QBM/2015/5/13/1572106333437952/1572106339598336/STEM/e26903dc9299421ba5484c8fd2cd39bf.png?resizew=13)
![](https://img.xkw.com/dksih/QBM/2015/5/13/1572106333437952/1572106339598336/STEM/0ca93da5b9664397818db69686d05e99.png?resizew=19)
![](https://img.xkw.com/dksih/QBM/2015/5/13/1572106333437952/1572106339598336/STEM/6749a8708afd4ba18cca1b6b6c5c9269.png?resizew=151)
![](https://img.xkw.com/dksih/QBM/2015/5/13/1572106333437952/1572106339598336/STEM/e182edb958c84fe9a0c9b10233845a91.png?resizew=36)
(1)试求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
(2)令
![](https://img.xkw.com/dksih/QBM/2015/5/13/1572106333437952/1572106339598336/STEM/ecb5020bf30d4b54968b6cced1146e2b.png?resizew=85)
![](https://img.xkw.com/dksih/QBM/2015/5/13/1572106333437952/1572106339598336/STEM/84407bb9caf34bbcbeb42ef37ea76a73.png?resizew=17)
![](https://img.xkw.com/dksih/QBM/2015/5/13/1572106333437952/1572106339598336/STEM/2d007bc2c06c4c85aa33872ce4a2ee78.png?resizew=29)
![](https://img.xkw.com/dksih/QBM/2015/5/13/1572106333437952/1572106339598336/STEM/e26903dc9299421ba5484c8fd2cd39bf.png?resizew=13)
![](https://img.xkw.com/dksih/QBM/2015/5/13/1572106333437952/1572106339598336/STEM/b25c025e9306451f844ddb0430a2bb2f.png?resizew=45)
(3)证明:对任意给定的
![](https://img.xkw.com/dksih/QBM/2015/5/13/1572106333437952/1572106339598336/STEM/0a6db1320cfa40bc92db6fe8aedef68f.png?resizew=76)
![](https://img.xkw.com/dksih/QBM/2015/5/13/1572106333437952/1572106339598336/STEM/dae2c6e3f84f4c2b8b81475bfe91016e.png?resizew=51)
![](https://img.xkw.com/dksih/QBM/2015/5/13/1572106333437952/1572106339598336/STEM/edc1899d2934421fa67ad8b7f509d5cc.png?resizew=41)
![](https://img.xkw.com/dksih/QBM/2015/5/13/1572106333437952/1572106339598336/STEM/2c1cd374b70d4c929a11b61cae221517.png?resizew=47)
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