名校
解题方法
1 . 在各项均不为零的数列
中,选取第
项、第
项,…,第
项,其中
,
.若新数列
为等比数列,则称新数列为
的一个长度为m的“等比子列”.已知等差数列
,其各项与公差d均不为零.
(1)若数列
满足
(
,
).请写出符合条件的所有等比子列;
(2)若
,数列
为
的一个长度为m的“等比子列”,其中
,公比为q,当q最小时,求
的通项公式;
(3)若公比为q的等比数列
,满足
,
,
(
,
),证明:数列
为数列
的“等比子列”.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27628b047da341c79074ea4aa938ddc8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/527093b2ec760913d0dccff8a099248b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0c5b45ef6860f96dd3f033b456056c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31ba164203399725ee3c6d42ba903b56.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(1)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9260f8989cfd0ffca5a49ffbc0668f14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1ba5610981c61a8022d945b395f813c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f94dbb638c192813faea2ae60882771a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31ba164203399725ee3c6d42ba903b56.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2eaa992a449b828df0ff545e233b279b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27628b047da341c79074ea4aa938ddc8.png)
(3)若公比为q的等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3aaee408bdec05bbdfcd4b841a331e61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/012f1e5df0528c0f9a5754b7dc84424e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac52d20d7bb3a6631f5035ef18b64c19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2afd8b154553478bc39b7cc7215aad7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed6af1644737a2948f30308a168ff07e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
您最近一年使用:0次
2 . 设,数列
满足
,数列
的通项公式为
.
(1)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcb148bd8ad91c0ef275571029eee56d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9eeeec6d9ce1bee1f657744582176cb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25c2a5f8ec179b72b201c3c0a670612a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.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/4c1a0517a786ca4f9d74691bef7d77b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.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/cadc27a59a738b9f6ae29fa7883754a3.png)
您最近一年使用:0次
2022-12-26更新
|
426次组卷
|
3卷引用:上海市格致中学2022-2023学年高二上学期12月月考数学试题
上海市格致中学2022-2023学年高二上学期12月月考数学试题上海市宝山区顾村中学2023-2024学年高二下学期3月阶段练习数学试题(已下线)拓展三:数列与不等式 -【帮课堂】2022-2023学年高二数学同步精品讲义(人教A版2019选择性必修第二册)
名校
3 . 无穷数列
,若存在正整数
,使得该数列由
个互不相同的实数组成,且对于任意的正整数
,
中至少有一个等于
,则称数列
具有性质
,集合![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc111429c53c669e45b3a37869b001eb.png)
(1)若
,
,判断数列
是否具有性质
;
(2)数列
具有性质
,且
,
,
,
,求
、
的值;
(3)数列
具有性质
,记集合
,将集合
中的所有元素按从小到大的顺序排列,得到数列
,记
,
,证明:若数列
具有性质
,则数列
是常数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/393cafa7e27934b2ed05d65af451f2fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cac190818c57f1504d04d86fea0c0204.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/393cafa7e27934b2ed05d65af451f2fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc111429c53c669e45b3a37869b001eb.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ec7510dc49bea69b5961573297f1289.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1505d56f0b35fe7f2de1fe1888036e4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
(2)数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb0abedd59476f8793d5a8948590a70c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/171a37e4d0bf1ef80a57e8349e8e3a6c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fdd9b5625cb5f752cd3481247a71b58.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e84c30444f13d37ada78285dc4f83b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46be9c9b05fd84ac9256d49a5a553af5.png)
(3)数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba62e5fc2a642f544a9eff5272f203d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11be0364247bc8af1552270971322971.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3315e9c6c2d3d60fb82ee0fa328fcbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1505d56f0b35fe7f2de1fe1888036e4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
您最近一年使用:0次
4 . 已知数列
中的相邻两项
,
是关于
的方程
的两个根,且
.
(1)求
,
,
,
;
(2)求数列
的前
项和
;
(3)记
,
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b7b6ecfdff9d2b29ef64d2a6f3343f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39773a450e3c30c72ead226d84e54563.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e17c3925955291056e16a4e075b3a13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23195d199724aea88a760a0ae35ff9b8.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f65fc200f10b97588a0c9896277c9c64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fd71bc7e6668f90f259ad0b06dd60c2.png)
(2)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2d51f9147b8265c0276c1f2c2659197.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b9a0d7150fb24be3e28ef7f0e18be93.png)
(3)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a742c7b44a3b6ebbbe78d5e0ad04bca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f0e01c1fac9f9ed8d588d4e85c0db8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c3e29cafd6334eca70149f61f34ca7c.png)
您最近一年使用:0次
2021-10-21更新
|
724次组卷
|
2卷引用:上海市复兴高级中学2021-2022学年高二上学期10月质量检测数学试题
名校
5 . 记无穷数列
的前
项中最大值为
,最小值为
,令
.
(1)若
,写出
,
,
,
的值;
(2)设
,若
,求
的值及
时数列
的前
项和
;
(3)求证:“数列
是等差数列”的充要条件是“数列
是等差数列”.
![](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/6ddad3d9fdb5e9951b6a1c31f9a72a71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0a625b91e0eba33b107550ee2a1e2f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/293570f1284f5161d0c9e83c1aef7777.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d207b73fd6b888db038e3e0d17383958.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b715e7842b95f654f16056a7c7f2abe9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13423c094861baf4b759b7f3d8c3c226.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57483e04fd1840c87ac5325157149877.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a548938d87c80ac47910607d3857007f.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd9a732c6e2fa8e55be1ba69627bb869.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5eda23dce727cb8d82fb27ec7db13bcb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5818ede14d21f6df9ef9c2bfe09286c.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/08eb71ecf8d733b6932f4680874dbbf3.png)
(3)求证:“数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fce83115a50f99e08e9a2db7267aeed.png)
您最近一年使用:0次
2019-04-28更新
|
672次组卷
|
4卷引用:上海市控江中学2022届高三上学期12月月考数学试题