1 . 已知无穷数列
,构造新数列
满足
,
满足
,
,
满足
,若
为常数数列,则称
为
阶等差数列;同理令
,
,
,
,若
为常数数列,则称
为
阶等比数列.
(1)已知
为二阶等差数列,且
,
,
,求
的通项公式;
(2)若
为
阶等差数列,
为一阶等比数列,证明:
为
阶等比数列;
(3)已知
,令
的前
项和为
,
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f22e75bf99dcb0fe32f66fa90a74f9a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dc06ff2f65c456efb6a114e67b62cb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d16e3ca4fb65342b0e9a0594d661ba3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eaf623a6dd661f87cf64ea150072fd78.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcbdf91bb5a49f293f9b52ac8dcfa35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdc2b6b23da3e065820c15cf6c675e20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ce1ca0ac3789432a3d1ce975a9a6f4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23ab5abd104649f9a3c231df9a55813a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ce1ca0ac3789432a3d1ce975a9a6f4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69c653f3a05dd25f6affdba7baeab38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f0606df3b6fbc6fcf863f9c59e0e54f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdc2b6b23da3e065820c15cf6c675e20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab214e5f34d976717284bed52c645e10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/020e796a0144ff9b3329c7064a01c5c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(1)已知
![](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/18d8e8f821111de8075e5c3dfb22a5d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56a4b318fa151fab14b3857942ed3cb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07418f1fa0f6569b50944b5a13ffa021.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b00f4eb7f1bd2ccefbabf0c1dfa8f69.png)
(3)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e7778648d93a10ed66ddd99672d3ff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a77316e06c00a9086be642f7f590684.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/a2a5a414f3f5d724058a2e887f3d1c0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c3fec47d2dd2b8099d86c87b6e57de8.png)
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2024·全国·模拟预测
2 . 甲、乙两名小朋友,每人手中各有3张龙年纪念卡片,其中甲手中的3张卡片为1张金色和2张银色,乙手中的3张卡片都是金色的,现在两人各从自己的卡片中随机取1张,去与对方交换,重复
次这样的操作,记甲手中银色纪念卡片
张,恰有2张银色纪念卡片的概率为
,恰有1张银色纪念卡片的概率为
.
(1)求
的值.
(2)问操作几次甲手中银色纪念卡片就可能首次出现0张,求首次出现这种情况的概率
.
(3)记
.
(i)证明数列
为等比数列,并求出
的通项公式.
(ii)求
的分布列及数学期望.(用
表示)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ffb021aa7d5a5c2f0691e337caad624.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ce603aa3abcb61750d2191aaa13dddc.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7f66b7e38f44f8cd5d48b3aa24a20fc.png)
(2)问操作几次甲手中银色纪念卡片就可能首次出现0张,求首次出现这种情况的概率
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
(3)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9131abf93295537bbc0c54a8c42e88e2.png)
(i)证明数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c895d4ce5ce82ef9b311b9369b4de11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(ii)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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3 . 在等比数列
中,
是函数
的两个极值点,若
,则t的值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd67f7444756faf766876de3fc6b1084.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fe34fddc0248037344564149a03a638.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f93c932af7de2f59447846c57fe52a0.png)
A.![]() | B.![]() | C.4 | D.5 |
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4 . 点列,就是将点的坐标按照一定关系进行排列.过曲线C:
上的点
作曲线C的切线
与曲线C交于
,过点
作曲线C的切线
与曲线C交于点
,依此类推,可得到点列:
,
,
,…,
,…,已知
.
(1)求数列
、
的通项公式;
(2)记点
到直线
(即直线
)的距离为
,求证:
;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c904567c3b3734e1eca8d042ef7a7b2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec391f08f1452fb3e0aebe7e12ba4fb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3a4422395ca20fe847419ec569e48b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b9cb8e6ff801523b0304576cd69fd2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eddbde5d269189fced4cc478908a6866.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec391f08f1452fb3e0aebe7e12ba4fb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3a4422395ca20fe847419ec569e48b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eddbde5d269189fced4cc478908a6866.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb9979465ce76b8582067703b39a0bc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87a60302649eb940748da818199e55da.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/086e9b14c35ef3c57b20f5e952ebf9c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36460040ddea4761eee10d537b14a1f6.png)
(2)记点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf83e20035c3afd6d26ebfd53d768a70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29a5b0f908cdae073db61be5b42fbcf7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c93f04dcabdafec74f98f4a1f4faa3fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82e260b088f071983f254ce8f5163fcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3db2da4189bf5f95ae10e6b96ee4b72e.png)
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5 . 已知“正项数列
满足
”,则“
”是“数列
为等比数列”的( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76a94ecbea274e2f765b0525b57831cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ccd2974f22054e744baf11dc860fb78.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
A.充分不必要条件 | B.必要不充分条件 | C.充要条件 | D.既不充分又不必要条件 |
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解题方法
6 .
的内角A,B,C的对边分别为a,b,c,且b是a,c的等比中项.
(1)求B的最大值:
(2)若C为钝角,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
(1)求B的最大值:
(2)若C为钝角,求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e188dce787593675ab005b13bfc64517.png)
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7 . 已知
为公差为2的等差数列
的前
项和,若数列
为等差数列.
(1)求
;
(2)求数列
的前
项和.
![](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/832fd7a51831135b6ee6a01981db250e.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
(2)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53c0adc9baa0a2bc60e4d8d89e03283c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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2024-04-05更新
|
1289次组卷
|
2卷引用:广东省深圳市东北师范大学附属中学深圳学校2024届高三下学期3月校内模拟测试数学试题
名校
解题方法
8 . 记
,
分别为数列
,
的前n项和.已知
为等比数列,
,
,
.
(1)求
,
的通项公式;
(2)求数列
的前2n项和.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](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/4dea1dd4ffcb4cf0697ca43079f6a1f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b38160d42f38e55c788965ab434f91f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f8643b494273cef85084829c99acbe0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f521a9a4126c66de71ca7305d29a6816.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec4bdc2a6d4fc387dc621f0b5a268c5.png)
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2024-03-10更新
|
989次组卷
|
3卷引用:2024届高三星云二月线上调研考试数学试题
2024届高三星云二月线上调研考试数学试题重庆市第一中学校2023-2024学年高二下学期开学考试数学试题(已下线)5.3.2 等比数列的前n项和(3知识点+8题型+强化训练)-【帮课堂】2023-2024学年高二数学同步学与练(人教B版2019选择性必修第三册)
名校
9 . 已知等差数列
的公差与等比数列
的公比相等,且
,
,
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/414187fca31df508dbf88d7f2bb83662.png)
______ ;若数列
和
的所有项合在一起,从小到大依次排列构成一个数列
,数列
的前
项和为
,则使得
成立的
的最小值为______ .
![](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/684b935a7274130d081bfa7b2b938023.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a89b8821c758c29c8b02bd79425ecbd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13624221ac7e06fc5ebd48b21ed0de10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/414187fca31df508dbf88d7f2bb83662.png)
![](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/57ef6d44448092ebdb9e4a49d866a749.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)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/055bd850286feebbeac5df481988b1d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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2024-03-08更新
|
1239次组卷
|
4卷引用:河北省部分学校联考2024届高三下学期3月模拟(二)数学试题
名校
解题方法
10 . 已知递增的等比数列
的前
项和为
,若
是
与
的等差中项,则
( )
![](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/a9adde0d99f886ea5c079b2eceeec93f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41a80da3456eae2c4d960fd167a50f5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc17ca3ab612ea9cf6cfa1eea53cb1eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef354e5c5ff828cc8d27c71badd40f98.png)
A.21 | B.21或57 | C.21或75 | D.57 |
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2024-03-07更新
|
1037次组卷
|
4卷引用:内蒙古赤峰市2023~2024学年高三上学期1.30模拟文科数学试题