1 . 设
的两个根分别为
,
,设
.
(1)求证:
;
(2)求
的个位数字.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50938cb46ccd303b2d0a9faf7791adf1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b12a4eecd249473a831d0ee472470240.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9565876bc50bceb63e5793c8c67a9032.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d216801533ab2a5e3107d750de0f54e.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a916f8905f8f60c36fbd8a61fd18cc70.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c523ff551d94b243e73e8e6b63a33d5.png)
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2 . 在数列
中,已知
,
.
(1)证明:
.
(2)证明:当
时,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2438f2272d7b7ab51dbbe587025a553d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc6b6bb6ebb130e774ae4331e8511131.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e011e6233ec4339fa324a51a21bc13f.png)
(2)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4996bf303bb42b9b36cf0b3ab46a3d0b.png)
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2023高三·全国·专题练习
3 . 设数列
满足
,
.
(1)证明:
.
(2)设数列
的前n项和为
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ea8d0e50065114b05ef2dc1ea1129cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb2dbc4c08aa70076c1c12daeedcb298.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c31f2f5b97bc76078c101082bb76bb6.png)
(2)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41cf1da18d91f7c98086553d157d1a87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ffb575d8e0365de6ccab0d0645fc78a.png)
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名校
解题方法
4 . 已知在
中,
.证明:
(1)
;
(2)
在
上恒成立;
(3)
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e92a98e220a9a1f2a1caa37e4cf4e213.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba5e4691210486a560c59df09937d9f8.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/991a6e773c41687e5b13d36da7612e01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d01dc2d99655cf7598837cb0886166ed.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb222ce13688da6fc57089ebf5812b0e.png)
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名校
5 . 如图,
是曲线
上的
个点,点
在
轴的正半轴上,且△
是正三角形
是坐标原点).
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/2/86311efd-f7c4-4a46-8bfb-5e6933f9905a.png?resizew=249)
(1)求
、
、
的值及数列
的递推公式;
(2)猜想点
的横坐标
关于
的表达式,并用数学归纳法证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/857d97c35c872937e4172dbc2ede1af2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/956620959792f8c3a28ed0fc4bf053d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3f94253ca8e33eca70b1a9ee6c2a0e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b27853b475a8bc2bfe297bbdbdaaa86d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a617f7ebacc3d380868cf6e002c0be3e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/2/86311efd-f7c4-4a46-8bfb-5e6933f9905a.png?resizew=249)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
(2)猜想点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b25286f6029da66ce5270aacd05184f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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名校
6 . 已知数列
,
,…,
的各项均为整数,且对任意的
,2,…,
,都有
.将
的所有项之和记为
.
(1)若
,
,求
的最大值;
(2)若
,求证:
;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/140b9dbcada4ac2e5fe3cc30009bcd67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5453ec6a9e8b96357c888ea863ddcf7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7334c46af837676ada9575630a48d60f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c45176df950dfe48b8ca7eac08ee349.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fa2648793a3889448088fa3f9f5aa49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/249ccefb918d906ca640ec76e53247e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bba7a40102e81f5759f7b05ebf5a18c.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebcc133d5b11b33a904875182d8c8261.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2815b24f5a89be7ae53aed93182e8988.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bba7a40102e81f5759f7b05ebf5a18c.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f8a0d4158a6df1bf0631095eb51c10c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5eb189e7f5f358b2de87dc4b93413366.png)
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2023高三·全国·专题练习
解题方法
7 . 已知数列
的前
项和为
,且
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
(1)证明:
是等比数列;
(2)求数列
的通项公式,并求出使得
成立的最小正整数n.(参考数据
)
![](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/1b8cb80269d538c2e30c0cf1c4884033.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c895d4ce5ce82ef9b311b9369b4de11.png)
(2)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/846fa57d92d6ad44d6a0cafad1e71ed4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef64850887c5c8cad4d574b0b09307a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8476e21ce074eb710ac64ad1eb630453.png)
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2023高三·全国·专题练习
8 . 设数列
满足
,证明:对任意的初值
,
存在并求此极限.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e976c0663fa749ca749f99842d21ca03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aba4c0a60d0a8b7d9702a25b5b324b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a460f28a8bf1bfe9d867828402bcccb.png)
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9 . 已知数列
的首项
,
,
.
(1)求
的通项公式;
(2)证明:对任意的
,
,
;
(3)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc6545b8eca1c4223ed701a199a85683.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7643e8b7aa32ebf299048417a94432dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42bab065e6362b760b5eec5b969c204c.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
(2)证明:对任意的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cd868bb05899fb3942c32cdb7d547b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7247567230a3bebb8fa497c2b22bb02.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78eca1f732051ae5f4fdfdb139c14672.png)
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