1 . 已知正项数列
中,
,前
项和为
,且______.请从下面两个条件中任选一个条件填在题目横线上,再作答.
条件:①
;②
.
(1)求数列
的通项公式;
(2)设
,数列
的前
项和为
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbaae3509b29f0bc77e8687702b7484d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8a0eecb5b800fce9ae10aed86ffee62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
条件:①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88ac30b318fc694a83797c3b7b759403.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aca5fa47a48f7f91593669b5bd7bb7e5.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c13f5e05ff277824a11dc48dcbae2d82.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/f9928e46511e601913619a427ded84a3.png)
您最近一年使用:0次
2024-01-16更新
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359次组卷
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2卷引用:新疆维吾尔自治区乌鲁木齐市第101中学2023-2024学年高三下学期4月月考数学试题
2 . 非零数列
满足
,且
.
(1)设
,证明:数列
是等差数列;
(2)设
,求
的前
项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b613d5800d61f4a25c7c739d680292dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cefeddf71dca8ae824328df3f0e5e1e.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fee3cf29d889864199a6db7b1685f179.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca857b7a6a1fe09827ecaa5f4c036069.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5ab0309e2cd35585ea9fb2cc3017abf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
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解题方法
3 . 已知公差不为0的等差数列
的首项
,设其前n项和为
,且
成等比数列.
(1)求
的通项公式及
;
(2)记
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13a6e1d671215fc96e4bee3541d1096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/970571e815c08e8d377b434eedfd72d3.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(2)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/032b74193c04dd5b9b389f93de59e2cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c215db1d8f69757118ad405b78035628.png)
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4 . 已知正项数列{an}的前n项和为Sn,在①
,且
;②
;③
,
,这三个条件中任选一个,解答下列问题:
(1)证明数列
是等比数列,并求其通项公式;
(2)已知
当
且
时,
,数列
的前n项和为
,若
恒成立,求
的最小值.
注:若选择不同的条件分别解答,则按第一个解答计分.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b2a09f14453f68329d983439a2ee3e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13a6e1d671215fc96e4bee3541d1096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3ff45e02dbec34a87856ab005ac0d02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e148ff4cabd6fb4d104d4611c670cae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13a6e1d671215fc96e4bee3541d1096.png)
(1)证明数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e0908c8a773fa043864160f4e69e057.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6dafd98e5b223908b13013c3cacc0386.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56cafe1878ccc6c84945fddc8609e40d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fafc28d435b6897b96b0daa8c0f9d850.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
注:若选择不同的条件分别解答,则按第一个解答计分.
您最近一年使用:0次
5 . 已知等差数列
的前
项和为
.
(1)求
及
;
(2)令
,求证:数列
的前
项和
.
![](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/09315cbd92484ce4edaff6bbe776e6fd.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(2)令
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/991c9c19c50f5879e81681affca5c26b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0791fc0d57d2e1b240c01d4c4901dadc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9f67984a336e63b128a8ccffaa6b0f8.png)
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|
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3卷引用:新疆新和县实验中学2023届高三上学期第一次月考数学(理)试题
名校
解题方法
6 . 设数列
的前
项和为
,已知
,
是公差为2的等差数列.
(1)求
的通项公式;
(2)设
,数列
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/b13a6e1d671215fc96e4bee3541d1096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dea1dd4ffcb4cf0697ca43079f6a1f2.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/216876de04325fd250c38c485cbc34b7.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/06143bd711d5af589ee94f419435788e.png)
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3卷引用:新疆巴音郭楞蒙古自治州若羌县中学2024届高三上学期6月摸底考后强化数学试题
新疆巴音郭楞蒙古自治州若羌县中学2024届高三上学期6月摸底考后强化数学试题湖北省黄冈市浠水县第一中学2023届高三下学期5月高考仿真模拟数学试题(已下线)专题11 数列前n项和的求法 微点3 裂项相消法求和(一)
名校
解题方法
7 . 已知数列
为公差不为零的等差数列,其前n项和为
,
,
.
(1)求
的通项公式
;
(2)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/34e3012b669194c74b11a95c18bbf667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a7f8f1b5bff849c18f368df4f6764ce.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e60fbec842e9f7b352d0ba3ad641946f.png)
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5卷引用:新疆乌鲁木齐市第十二中学2024届高三下学期5月月考数学试题
新疆乌鲁木齐市第十二中学2024届高三下学期5月月考数学试题安徽省合肥市2023届高三下学期第一次教学质量检测数学试题(已下线)模块九 数列-1(已下线)专题10数列(解答题)江西省南昌市第十九中学2022-2023学年高二下学期期末考试数学试卷
解题方法
8 . 已知非零数列
的前n项和为
,且满足
,其中p为常数,且
.
(1)证明:数列
是等比数列;
(2)若
,
,数列
的前n项和为
,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/b7a48c92685da35053d50b643b0ee77f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dea7e0ac16c02bd211e9926c44e50334.png)
(1)证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7acff98078cdd32804d8f1c4efbe2ddd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e33761fb26db9a36416a503fa7f0ed7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9928e46511e601913619a427ded84a3.png)
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2卷引用:新疆乌鲁木齐市等5地2023届高三高考第二次适应性检测数学(理)试题
9 . 设数列
的前n项积为
,且
.
(1)求证数列
是等差数列;
(2)设
,求数列
的前n项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f5a629a31ee190a2eddf393425813a3.png)
(1)求证数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a40442811c08c432ec613102e4502c0.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/125283c86155f0483dc68c92baa0b52a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
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4卷引用:新疆乌鲁木齐市第八中学2021-2022学年高二下学期期中考试数学(理)试题
新疆乌鲁木齐市第八中学2021-2022学年高二下学期期中考试数学(理)试题陕西省西安工业大学附属中学2022届高三下学期第七次适应性训练文科数学试题(已下线)4.4 求和方法(精练)-【一隅三反】2023年高考数学一轮复习(基础版)(新高考地区专用)福建省龙岩市上杭县才溪中学2023-2024学年高二上学期9月月考数学试题
10 . 已知数列
满足:
.
(1)证明数列
为等差数列,并求数列
的通项公式.
(2)若
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d41bb2f7a5354c6dbaa4f51a25be9ff.png)
(1)证明数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d82c65a855b1eed9c43e6829f6c3bffb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e57d48e6f3765cde24016384bbc73be8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b2c37e887cb17f4cb8b4933d297df8e.png)
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