题目
题型:不详难度:来源:
已知数列
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012049-89748.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012051-79632.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012051-78675.png)
(1)当
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012051-70981.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012051-65210.png)
(2)在(1)的条件下,若数列
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012051-79382.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012052-14431.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012052-16489.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012052-87445.png)
(3)对于
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012052-18105.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012051-65210.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012053-16491.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012053-91267.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012053-28329.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012053-36461.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012054-34931.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012054-45077.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012053-28329.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012053-36461.png)
解析
试题分析:(I) 当
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012051-70981.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012054-26582.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012054-76756.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012055-83039.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012051-65210.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012055-36446.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012055-47387.png)
(II)先求出当
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012055-12028.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012056-20400.png)
因为b1=1也满足上式,因而当
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012056-75915.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012056-14819.png)
然后根据
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012056-64254.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012054-45077.png)
(3) 由
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012056-61078.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012057-59293.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012057-48509.png)
即
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012057-41387.png)
从而得到
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012057-50832.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012057-38258.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012058-21019.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012058-29508.png)
然后可得
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012058-99240.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012058-76230.png)
通过分组求和即可求出Sn,到此问题基本得以解决.
(1)当
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012051-70981.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012054-26582.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012054-76756.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012059-98891.png)
故数列
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012051-65210.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012055-36446.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012055-47387.png)
故数列
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012051-65210.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012054-34931.png)
(2)由(1)得,
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012059-13672.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012059-82719.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012055-12028.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012059-86337.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012100-21777.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012100-89210.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012056-75915.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012056-14819.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012100-77605.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012056-75915.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012100-40106.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012101-25672.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012101-12536.png)
(3)解法一:由
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012056-61078.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012057-59293.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012057-48509.png)
即
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012057-41387.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012101-75184.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012057-38258.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012058-21019.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012101-42255.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012102-79292.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012102-58223.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012102-68701.png)
因此,
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012053-28329.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012053-36461.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012102-26679.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012102-60251.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012053-36461.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012103-79101.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012103-92595.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012103-65071.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012059-82719.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012053-28329.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012053-36461.png)
解法二:同解法一得
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012103-43356.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012103-53673.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012102-79292.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012102-58223.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012104-24364.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012104-81866.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012104-85060.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012104-65197.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012059-82719.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012053-28329.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012053-36461.png)
点评:(1)等差等比数列的定义是判定一个数列是否是等差或等比数列的依据,要勿必掌握.(2)三角函数公式的变形也是解决本题的基础,因此要熟记常见的变形公式如:
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012105-20124.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925012105-96079.png)
(3)在比较两个数或式子大小不易直接比较时,作差比较法是常用也是很有效的方法之一.
核心考点
试题【(本小题满分14分)已知数列满足:其中(1)当时,求的通项公式;(2)在(1)的条件下,若数列中,且求证:对于恒成立;(3)对于设的前项和为,试比较与的大小.】;主要考察你对两角和与差的三角函数等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925010905-39263.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925010905-38620.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925010905-38620.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925010905-38620.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925010256-78067.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925010257-56321.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925010257-23514.png)
A.-3 | B.-1 | C.1 | D.3 |
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925002827-45077.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925002145-62900.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925002145-49697.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925002145-76034.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925001038-71056.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925001039-15029.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190925/20190925001039-57043.png)
A.![]() | B.![]() | C.1 | D.![]() |
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