名校
解题方法
1 . 设数列
的前n项和为
,且
,
.
(1)求数列
的通项公式:
(2)设数列
的前n项和为
,求证:
为定值;
(3)判断数列
中是否存在三项成等差数列,并证明你的结论.
![](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/c8370a173854471a3eb27637993a3d5d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/362832fa3d3c13c1eafd565349d66dce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b87635913b4f90a784edd6ef79f2aec.png)
(3)判断数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf6421b801b00bceab7547d9ed86874e.png)
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解题方法
2 . 已知
是常数,在数列
中,
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c94ae936f82425aadd82bfbc76079d2a.png)
(1)若
,求
的值;
(2)若
=4,证明:数列
是等比数列,并求数列
的通项公式;
(3)在(2)的条件下,设数列
的前
项和为
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039e4fe671d61e59b96ee525c9df43e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c94ae936f82425aadd82bfbc76079d2a.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15a783088120d67cc98936081e80fb7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e2de706dc5f0439b989273a5367f63a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)在(2)的条件下,设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41cf1da18d91f7c98086553d157d1a87.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/1302abaebc9df026c2a83291063e83b4.png)
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3 . 定义:对于一个项数为
的数列
,若存在
且
,使得数列
的前k项和与剩下项的和相等(若仅为1项,则和为该项本身),我们称该数列是“等和数列”.例如:因为
,所以数列3,2,1是“等和数列”.请解答以下问题:
(1)数列1,2,p,4是“等和数列”,求实数p的值;
(2)项数为
的等差数列
的前n项和为
,
,求证:
是“等和数列”.
(3)
是公比为q项数为
的等比数列
,其中
且
恒成立.判断
是不是“等和数列”,并证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adb8ed53d63acbc84f971bf27e03b854.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e36bff57bcfa86432b340e25e51d42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfae43a1ea1f5f3a031d224ac61a5f37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0df3727c94ac7dd40a1167fbefe77555.png)
(1)数列1,2,p,4是“等和数列”,求实数p的值;
(2)项数为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/084560d2f7832c82cea659b4be76bf8b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82078fcb4aeffb2fb56924507739f33e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a59cafa2aa63beea36dec11d61a1ddb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26327328443a6e65db56cf4d1254b011.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/699e55c211a6e091cc7a9d2cde3ed981.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a38c773b8a5cccaf44ccc24e4fb7cbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
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4 . 已知数列
满足
,
,
.
(1)求证数列
是等比数列,并求数列
的通项公式;
(2)设
,数列
的前
项和为
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13a6e1d671215fc96e4bee3541d1096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3369ae2337f8d6a049fd8e5a9f313f87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
(1)求证数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c895d4ce5ce82ef9b311b9369b4de11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/017a49f5aebadd0aef6449abb44b8baa.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/ab2a823b63d3f6634f0a294bec20aea0.png)
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5 . 设集合
由满足下列两个条件的数列
构成:①
②存在实数
使得
对任意正整数
都成立.
(1)现在给出只有5项的有限数列
试判断数列
是否为集合
的元素;
(2)设数列
的前项和为
且
若对任意正整数
点
均在直线
上,证明:数列
并写出实数
的取值范围;
(3)设数列
若数列
没有最大值,求证:数列
一定是单调递增数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91edc7e2d4811f5ea6c01284cf00393a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbb396d39062b87507b6d9ce28b841e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/663a61ad241d5d874c9a9362f0ee917c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f96ecb5f9618e6488e0459c8efcd00c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(1)现在给出只有5项的有限数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74eb4ff1f7cecdbe5b9ad2f87a6d869e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91edc7e2d4811f5ea6c01284cf00393a.png)
(2)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/117464f527849ab995858aaa20f4175b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/723a219f613878364e4a6ac272055099.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa38dba3681c7b18114885e82463880a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3da02b21ba6425f0326d26a16a15c091.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9450684ca4c5f6ee21f226f5f40abe2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c80de4ac58be8a7e5486852cf5249891.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/663a61ad241d5d874c9a9362f0ee917c.png)
(3)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/112be839f8f70a099939d59e1450861a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
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解题方法
6 . 已知各项为正的数列
满足:
,
(
).
(1)求
;
(2)证明:
(
);
(3)记数列
的前
项和为
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/135610aed76b3236cdaf3931481556f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e0ff8ff5b51d663c040810957242ba9.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed529240a883f68f0921e818addeb9c8.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7eeb650c5d035f8f67a65788f7c1ae67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e0ff8ff5b51d663c040810957242ba9.png)
(3)记数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a15c37e6c1c27103628017944193e75e.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/c9f950fedce1e5ead461e7f52b734908.png)
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名校
7 . 设
是数列
的前
项之积,且满足
,
.
(1)求证:数列
是等比数列,并写出数列
的通项公式;
(2)设
是数列
是前
项之和,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](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/06329900217818d85b674673058bfa59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
(1)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc3c98f63dd4f25ec5b766727e33f1b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](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/ac573ecf9341af2d9b95c4d40a6c0072.png)
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名校
解题方法
8 . 已知函数
.
(1)若
,求
在点
处的切线方程;
(2)令
,判断
在
上极值点的个数,并加以证明;
(3)令
,定义数列
. 当
且
时,求证:对于任意的
,恒有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffb46a163897842c1fe507d8fca253bc.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e65397f11ea8af736f38debadf420c4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bea9227dd0104da58e0c40952cc87ed.png)
(2)令
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fa80a12751a326ffabe3115ff779983.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/870ebc2f7aabb028024894568d749934.png)
(3)令
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0138235d884d203490107787ea2e2830.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff61248b4e57f4f9f5b2d9ab74a82ae8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e65397f11ea8af736f38debadf420c4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a06b9066e5f16f7a4c2ccc88d48fbea4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fee9e71a4c714607f4b7af44337c411.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ad94568bce23439be8ded967d870c98.png)
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名校
9 . 已知数列
满足上:
,
.
(1)若
,证明:数列
是等差数列;
(2)若
,判断数列
的单调性并说明理由;
(3)若
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b66280badd3b6c4cc7629f024d1e67b.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a3c442579603164f3fc19458677d307.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87743e3348c037162aa605bb6bb2220c.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b86304c3e26200299a0480641525a283.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edcf33b2a94eae16760d746f9b4b8dbc.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b86304c3e26200299a0480641525a283.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d27884f5e62dda8dd97ecb5d62f86a0.png)
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10 . 已知数列
满足
,
,
,又
.
(Ⅰ)求证数列
是等比数列,并求出
的通项公式;
(Ⅱ)若
的前
和为
,
.
①判断并证明数列
的单调性;
②求证:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef56bd576cd62491ff07d614a82bb517.png)
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d4ac6110d24aef45673a9934c664f8e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2693734765399876e9e93cdb110231c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5fd5eb05c94e2a151dde58095a5f3e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa9cf7a9f015adee770033626528eb65.png)
(Ⅰ)求证数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d82c65a855b1eed9c43e6829f6c3bffb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d4ac6110d24aef45673a9934c664f8e.png)
(Ⅱ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fce83115a50f99e08e9a2db7267aeed.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/cc8bb07c3ef0cf84181557b568ebfbd9.png)
①判断并证明数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5eded65284816fdf6bf335b0c2a78e6a.png)
②求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef56bd576cd62491ff07d614a82bb517.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69d307ec71820b6536453fbdb5069da3.png)
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