名校
解题方法
1 . 已知数列
是各项均不为0的等差数列,
为其前n项和,且满足
,
,数列
的前n项和为
.
(1)求数列
的通项公式及数列
的前n项和
.
(2)是否存在正整数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/280860dd039e1305a5ccc455f63e8223.png)
,使得
,
,
成等比数列?若存在,求出所有的m,n的值;若不存在,请说明理由.
![](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/cfb0594573b831a8b5693d993b322804.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d005409790b3192705a181b2c8e7dfed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
(1)求数列
![](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/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
(2)是否存在正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/280860dd039e1305a5ccc455f63e8223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c949da72545619b5c328dd250bda0ee7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e9a724b59c890095baa5cb73e267c44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7e74be91bfe4bc209da7539dbf9b72c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
您最近一年使用:0次
2020-02-20更新
|
843次组卷
|
4卷引用:上海市进才中学2023届高三上学期11月月考数学试题
上海市进才中学2023届高三上学期11月月考数学试题(已下线)专题17 数列探索型、存在型问题的解法 微点3 数列探索型、存在型问题综合训练浙江省宁波市鄞州中学2018-2019学年高一下学期期中数学试题江西省吉安市永丰县永丰中学2022-2023学年高二下学期期中考试数学试题
名校
2 . 由9个正数组成的矩阵
中,每行中的三个数成等差数列,且
、
、
成等比数列,下列四个判断正确的有( )
①第2列
,
,
必成等比数列②第1列
,
,
不一定成等比数列
③
④若9个数之和等于9,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56952d18c393108434a65bc835cd7621.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d0639c2677e726834fda9f48c847424.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eeb9d1aa1cc02e6fe13141ecf5f2549b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bab5310774cd482fffd19586983b44c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/699f39dd97b907f10708e0dc75076f8a.png)
①第2列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e76d1d8e50dda4d50229a8a20c57e58.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33e11a5b70e1e2e685d1783a4707872e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/498d357985304a737e7a01272ebbcf15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e84c30444f13d37ada78285dc4f83b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc29ee719feeedfbc8c529cf11348abf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3a555e932cee7a505419ca454e4d1a7.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b829cb8d1278c92e5976b0ab4716bdd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56952d18c393108434a65bc835cd7621.png)
A.3个 | B.2个 | C.1个 | D.0个 |
您最近一年使用:0次
解题方法
3 . 设数列
的前
项和为
,对于任意的
,都有
.
(1)求数列
的首项
及数列的递推关系式
;
(2)若数列
成等比数列,求常数
的值,并求数列
的通项公式;
(3)数列
中是否存在三项
、
、
,它们组成等差数列?若存在,请求出一组适合条件的项;若不存在,请说明理由.
![](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/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b737ea1ec08b8c0da99dcfd6beea4560.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cfb19f0c37a72b33083ae9319f11a74.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/548c93a50c4ec17820f70d07358f6ac0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a93301f9e8004bf6c59804e1ae601bfa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de009d9df65374c870a4012cf5db28df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69790350b5ff13d020d7db465d13dc5.png)
您最近一年使用:0次
2020-02-09更新
|
613次组卷
|
2卷引用:2016届上海市嘉定区高三第三次模拟练习(文)数学试题
解题方法
4 . 已知数列
中,
,前
项和为
,且
.
(1)求
,
的值;
(2)证明:数列
是等差数列,并写出其通项公式;
(3)设
(
),试问是否存在正整数
,
(其中
,使得
,
,
成等比数列?若存在,求出所有满足条件的数对
;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f966272f7781790ff27e40db6b525253.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/e8b15fd3a64ad5b695a1a4b0e4225466.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
(2)证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ad8131d0b2e2e796a52d12a7ab5a74b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45a5fdd752754e29482eab73760a1704.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b715e7842b95f654f16056a7c7f2abe9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d7dd877103f225767609289fc7a25f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87cf58a39b00433d2ffbf34e86ca2f36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3abe178f96b431cf46aa8f394108d0d3.png)
您最近一年使用:0次
名校
5 . 设数列
,
满足
,
,
,且数列
是等差数列,数列
是等比数列.
(1)求数列
和
的通项公式;
(2)是否存在
,使
,若存在,求出
,若不存在,说明理由.
![](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/1bb88d8831173a3319d95c502110ab31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8592b997d19abf462dfa657056dea220.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f9a9ec3de6a5e79e9224456ef761212.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f05946117b44701da227291e10c9f40f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02a7513d87a7458d879211a14c59ec2e.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)是否存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bc1efe01d419af89e83ea54b5679b81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/369111b066c43eade67ac7ffbead2c47.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
您最近一年使用:0次
2020-02-05更新
|
225次组卷
|
3卷引用:2017届上海市上海中学高考模拟试卷(4)数学试题
名校
6 . 已知数列
是公差
的等差数列,且
.
(1)求
的前
项的和
;
(2)若
,问在数列
中是否存在一项
(
是正整数),使得
成等比数列,若存在,求出
的值,若不存在,请说明理由;
(3)若存在自然数
(
是正整数),满足
,使得
成等比数列,求所有整数
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/812be9806122241c476ba1db516c4823.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b83d73d014a0ca4aff4282228312f55.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8d02ea8c4988c5c28ab93f0d70fb55a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05201ef79a5d5904f492845396fb5470.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ced4e381e8c3336848b8c436dbc584f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/681ae1522a36768618f7ddaf74abbb7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/752a364c38482bcfa84d8d8a35fa06e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)若存在自然数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcc460dd6699e88e61cd6e230fc7fcf0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65edac3f3fee652a86f2a2f3b3b51fbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/517b00483a3b651e222c893ad8e6a14d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
您最近一年使用:0次
名校
7 . 设
是公比为
的等比数列,首项
,对于
,
,当且仅当
时,数列
的前
项和取得最大值,则
的取值范围为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa08a089a48737b6a0c47cd150b9f12a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/680018f55067af9d7fa8bbd9bf6233d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fac3649308b528fd56545ba102dc42d5.png)
![](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/9aa8a716a31b0f51b70fdf9bdb257909.png)
A.![]() | B.![]() | C.![]() | D.![]() |
您最近一年使用:0次
2020-01-13更新
|
561次组卷
|
2卷引用:上海市延安中学2016-2017学年高三下学期开学考试数学试题
名校
8 . 已知等比数列
的公比为
,它的前
项积为
,且满足
,
,
,给出以下四个命题:①
;②
;③
为
的最大值;④ 使
成立的最大的正整数
为4031;则其中正确命题的序号为________
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.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/8141d87fb02b08c88b0c9f27f839a7d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8afceb7e30c39a68170e0a8283050c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64e8696b37c79582b9c055cd972a65ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eda6dc559d07bc22c9a0ed1e3a6d01d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c938b8696465cf9b29646512c56874ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b3f92878c2f19d2825a6b5d4f9d884e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6ae93e401b499b0e39f251279b5663c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
您最近一年使用:0次
2020-01-08更新
|
687次组卷
|
4卷引用:上海市六校2016届高三下学期3月综合素养调研(理)数学试题
名校
9 . 已知数列
满足
,
,其中
是等差数列,且
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fc588e26774778d4045118bb633219a.png)
________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0914c295f572c98dd043d4f84268934.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/010baf415f792018ad9abd752e37b983.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f78927a4313925a218f7cf271c2c7a32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/546b3ee831eb68e290181418d0a175e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fc588e26774778d4045118bb633219a.png)
您最近一年使用:0次
2020-01-07更新
|
480次组卷
|
4卷引用:上海市闵行区2017-2018学年高三上学期期末质量调研数学试题
10 . 已知无穷数列
的各项都是正数,其前
项和为
,且满足:
,
,其中
,常数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
.
(1)求证:
是一个定值;
(2)若数列
是一个周期数列(存在正整数
,使得对任意
,都有
成立,则称
为周期数列,
为它的一个周期),求该数列的最小周期;
(3)若数列
是各项均为有理数的等差数列,
(
),问:数列
中的所有项是否都是数列
中的项?若是,请说明理由;若不是,请举出反例.
![](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/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7fab51121848ce166035ceab6f4e00b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf2ad613ce3380f983f09c8690e7df4b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c400a615a16a1662de98dfb4e49d58d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fe80aeedc8536a87704740c14c73acb.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d87e872b54bef7a4843c26c1f734fa7f.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b2b94cbf8f1acc77ed2618d9ba5756a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
(3)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c77a0d9091238b8d5eab4e4a8129149.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38ef4c4439b36c2847b0056a116d56d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
您最近一年使用:0次
2020-01-07更新
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353次组卷
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3卷引用:2017年上海市长宁、嘉定区高三上学期期末质量调研(一模)数学试题