1 . 已知
为等差数列,
为公比
的等比数列,且
,
,
.
(1)求
与
的通项公式;
(2)设
,求数列
的前
项和
;
![](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/45482d31d1d7448c9f3922b4d2a55331.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebaf2a2590bb84d646957f913d78f6dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/769fe52ac96348d3b12d23d06d702595.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccfe26b3ab1cd6a93075f24b696b0cef.png)
(1)求
![](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/f56b8ba45240095a2a2a36cc4b2ad180.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a107ab533d39b3bd0933429156ff33bb.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
2 . 在密码学领域,欧拉函数是非常重要的,其中最著名的应用就是在RSA加密算法中的应用.设p,q是两个正整数,若p,q的最大公约数是1,则称p,q互素.对于任意正整数n,欧拉函数是不超过n且与n互素的正整数的个数,记为
.
(1)试求
,
,
,
的值;
(2)设n是一个正整数,p,q是两个不同的素数.试求
,
与φ(p)和φ(q)的关系;
(3)RSA算法是一种非对称加密算法,它使用了两个不同的密钥:公钥和私钥.具体而言:
①准备两个不同的、足够大的素数p,q;
②计算
,欧拉函数
;
③求正整数k,使得kq除以
的余数是1;
④其中
称为公钥,
称为私钥.
已知计算机工程师在某RSA加密算法中公布的公钥是
.若满足题意的正整数k从小到大排列得到一列数记为数列
,数列
满足
,求数列
的前n项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbc89a53c03cb86fb653bb82128f6cba.png)
(1)试求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ccc57e5668f2a2c1cbc078a767b6855.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00f51f00d1a8a2f57f9e91d1f0264361.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c48460227f1fa924963cbc7878335152.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31457b25a8c32ecb910058736b337a49.png)
(2)设n是一个正整数,p,q是两个不同的素数.试求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0c104e0377b841ff77ab48b63e90470.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/647a247eba3658ab991c7f88f877f3b1.png)
(3)RSA算法是一种非对称加密算法,它使用了两个不同的密钥:公钥和私钥.具体而言:
①准备两个不同的、足够大的素数p,q;
②计算
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/617a64377b9f00c58ebe10841c402e32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbc89a53c03cb86fb653bb82128f6cba.png)
③求正整数k,使得kq除以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbc89a53c03cb86fb653bb82128f6cba.png)
④其中
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fee0c3b8386825011b6f2b74f18069a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ceb955cff0a243b938fe2d2d1e8a5dc.png)
已知计算机工程师在某RSA加密算法中公布的公钥是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5880d2e3f1a34188bf67a29e8de52f99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41b124fdd8b8097233e3d15417a779f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d9b38646bc714b68d44b7c954e7f4c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
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4卷引用:福建省厦门双十中学2023-2024学年高二下学期第一次月考数学试题
3 . 已知数列
的前
项和
,数列
满足:
.
(1)证明:
是等比数列;
(2)设数列
的前
项和为
,且
,求
;
(3)设数列
满足:
.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/5886e031a95a8d52c9306e6b1c518abc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65f2ecc6870129d1b5fa7f97b0824b83.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/921439ba032dd3fdec48755411b04533.png)
(2)设数列
![](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/18ec3b51bbda2de5b7a2e0360c8adc46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
(3)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8eb0aeb50edc4bfa079dc925aade88f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2cbe03ddf8f76a8d983ad63277ea2a3.png)
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福建省莆田第二中学2023-2024学年高二下学期3月月考数学试卷福建省福州第一中学2023-2024学年高二上学期第二学段模块考试数学试卷(已下线)江苏省南通市2024届高三第二次调研测试数学试题变式题 16-19(已下线)江苏省泰州市2024届高三第二次调研测试数学试题变式题16-19
4 . 已知数列
中,
,
,
.
(1)求数列
的通项公式;
(2)设
,求数列
的前
项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18832a9bc8edc29e02a9e03768eed287.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cca1d86c9f078347773f700fee49d1d8.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/114f1232f0e651086335404e8cf45651.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
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5 . 已知等差数列
满足
,
,公比不为
的等比数列
满足
,
.
(1)求
与
通项公式;
(2)设
,求
的前
项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18d8e8f821111de8075e5c3dfb22a5d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1d7127a0c69478f357f84de6241ae43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acbc6a613224461ade69362d46550474.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96b7b8f45c4af23b598b1bfee01db679.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f152df374ac70a9b006253fb89e56fb8.png)
(1)求
![](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/3897eb4c2e5b41e8b9b9641eb07f197e.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/08eb71ecf8d733b6932f4680874dbbf3.png)
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福建省福州市长乐第一中学2024届高三上学期1月考试数学试题湖南省张家界市民族中学2023-2024学年高二上学期第四次月考数学试题山东省泰安市泰山外国语学校2024届高三上学期期末数学试题(已下线)专题04 数列通项与求和技巧总结(十大考点)-【寒假自学课】2024年高二数学寒假提升学与练(人教A版2019)(已下线)考点10 数列求和 2024届高考数学考点总动员【练】
名校
解题方法
6 . 已知正项数列
满足:对任意正整数
,都有
成等差数列,
成等比数列,且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e719bd380ea9dd3ec2a20242bdf6380.png)
(1)求证:数列
是等差数列;
(2)设数列
的前项和为
,如果对任意正整数
,不等式
恒成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0197eeeeaafec6b1fdd7bb8509572f6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a20dfd9c14d834c66b2070c41f66eee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5b55761181faa05961286eedfebca4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e719bd380ea9dd3ec2a20242bdf6380.png)
(1)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce4f93dca4192c87d1ac77a2456bf12e.png)
(2)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b01041691ad489f126f05c18ea8f0fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfd0dc83494c84b81687cf8c38736b8b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
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3卷引用:福建省泉州市第一中学2024届高三上学期12月月考数学试题
7 . 在数列
中,
,
的前
项为
.
(1)求证:
为等差数列,并求
的通项公式;
(2)当
时,
恒成立,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ea8d0e50065114b05ef2dc1ea1129cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97a0a484cf87cb3bd96c3db9736c6f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62aadbfe3ef08851f220c3371684a1bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bb34ab1175fd4f7a8336221e559a784.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
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名校
解题方法
8 . 已知数列
的前n项和为
,
,且
.
(1)求证:数列
为等差数列;
(2)已知等差数列
满足
,其前9项和为63.令
,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/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc726d4282585efb91f8c34f5fd5cead.png)
(1)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dea1dd4ffcb4cf0697ca43079f6a1f2.png)
(2)已知等差数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77d9bd40057948c5e3eb23064a673284.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/347bbb4dc9eaf978094e8bb89d41c56d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5608d0c8d3b5b997012cb6dc698d9f4.png)
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名校
解题方法
9 . 在
平面上有一系列点
,对每个正整数
,点
位于函数
的图象上,以点
为圆心的
都与
轴相切,且
与
外切.若
,且
的前
项之和为
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bc625e19e7ca2b9d097f67a3d472e47.png)
__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1552c675b51d9d69cd32e061f8420bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf83e20035c3afd6d26ebfd53d768a70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/593f58c07b92d80b65f71e91b9991c04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf83e20035c3afd6d26ebfd53d768a70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b572c2c8262bb7bf6a8c9cdf1ebead22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b572c2c8262bb7bf6a8c9cdf1ebead22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d83258693b38108f4899207752b2e38e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87a60302649eb940748da818199e55da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f213202ccda6d7e6dcf21611e81c7b3.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/3bc625e19e7ca2b9d097f67a3d472e47.png)
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名校
解题方法
10 . 设
为数列
的前
项和,
.
(1)求数列
的通项公式;
(2)设
,证明:
.
![](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/eb4487d7daca378b322a42a8d04f341b.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3f770c2751f5f81c9b4419e4e99d1f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a051cd30dd080d1a1a22b46b6444ae9.png)
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8卷引用:福建省百校联考2024届高三上学期12月月考数学试题
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