【题文】,是方程的两根, 数列是公差为正的等差数列,数列的前项和为,且.(1)求数列,的通项公式;(2)记=,求数列的前项和.
题型:难度:来源:
【题文】
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162938-91999.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162938-41615.png)
是方程
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162939-11818.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162939-47289.png)
的两根, 数列
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162939-65792.png)
是公差为正的等差数列,数列
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162939-17966.png)
的前
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162940-98518.png)
项和为
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162940-17951.png)
,且
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162940-17951.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162940-19978.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162940-18267.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162941-45707.png)
.
(1)求数列
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162939-65792.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162939-17966.png)
的通项公式;
(2)记
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162941-67018.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162941-54193.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162940-18267.png)
,求数列
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162941-40360.png)
的前
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162940-98518.png)
项和
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162942-24445.png)
.
答案
解析
【解析】
试题分析:解:(1)由
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162943-73486.png)
.且
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162943-36956.png)
得
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162943-78096.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162944-56956.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162944-27560.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162942-35816.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162944-51239.png)
中,令
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162945-36108.png)
得
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162945-21864.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162945-76867.png)
时,T
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162946-48381.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162946-67358.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162946-65801.png)
,
两式相减得
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162946-12224.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162942-97169.png)
.
(2)
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162947-97417.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162947-14043.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162948-96647.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162948-63209.png)
=2
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162948-22010.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162948-95166.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162943-39787.png)
考点:等差数列和数列的求和
点评:解决该试题的关键是能利用等差数列的连个基本量表示得到其通项公式,同时得到两个数列的通项公式,同时结合错位相减法来求和,属于中档题。
举一反三
【题文】若关于
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162920-10350.png)
的方程
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162920-50006.png)
有四个不相等的实根,则实数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162920-63433.png)
的取值范围是( )
【题文】函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162909-35776.png)
的零点的个数是
.
【题文】函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162823-29445.png)
的零点所在的区间是( )
【题文】函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162749-35975.png)
的零点必落在区间 ( )
【题文】方程
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162740-18112.png)
所表示的曲线图形是( )
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327162740-30591.jpg)
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