题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010085411-40008.png)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010085411-26558.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010085411-14096.png)
(1)求证:数列
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010085411-66826.png)
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010085412-27895.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010085412-73025.png)
答案
解析
(1)利用f (x)=mx(m为常数,m>0且m≠1).代入an,求出an的表达式,利用等差数列的定义,证明数列{an}是等差数列;
(2)通过bn=an f (an),且数列{bn}的前n项和为Sn,当m=2时,求出Sn的表达式,利用错位相减法求出Sn;
解:(1)由题意f(an)=
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010085412-59767.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010085412-73567.png)
∴an=n+1,(2分) ∴an+1-an=1,
∴数列{an}是以2为首项,1为公差的等差数列.
(2)由题意
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010085412-27895.png)
当m=2时,bn=(n+1)·2n+1
∴Sn=2·22+3·23+4·24+…+(n+1)·2n+1 ①
①式两端同乘以2,得
2Sn=2·23+3·24+4·25+…+n·2n+1+(n+1)·2n+2 ②
②-①并整理,得
Sn=-2·22-23-24-25-…-2n+1+(n+1)·2n+2
=-22-(22+23+24+…+2n+1)+(n+1)·2n+2
=-22-
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010085413-62423.png)
=-22+22(1-2n)+(n+1)·2n+2=2n+2·n.
核心考点
试题【已知(m为常数,m>0且m≠1).设(n∈)是首项为m2,公比为m的等比数列.(1)求证:数列是等差数列; (2)若,且数列的前n项和为Sn,当m=2时】;主要考察你对等差数列等知识点的理解。[详细]
举一反三
A.等比数列 | B.等差数列 |
C.既是等差又是等比数列 | D.既不是等差又不是等比数列 |
(2,3),(3,2),(4,1),(1,5),(2,4),
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010085023-89257.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010085023-89257.png)
A.(3,8) | B.(4,7) | C.(4,8) | D.(5,7) |
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010084847-19202.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010084847-10399.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010084847-92409.png)
(1)设bn=
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010084847-10399.png)
(2)求数列{an}的前n项和Sn.
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010084824-37723.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010084825-68595.png)
(Ⅰ)若
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010084824-37723.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010084824-37723.png)
(Ⅱ)对于(Ⅰ)中
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010084824-37723.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010084825-50296.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010084825-99066.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010084825-46005.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010084826-31367.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010084730-53599.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010084730-47794.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010084730-58839.png)
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