【题文】已知函数f(x)满足f(x+y)=f(x)·f(y),且f(1)=. (1)当n∈N*时,求f(n)的表达式;(2)当an=n·f(n), n∈N*,求
题型:难度:来源:
【题文】已知函数f(x)满足f(x+y)=f(x)·f(y),且f(1)=
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325113005-51931.gif)
.
(1)当n∈N
*时,求f(n)的表达式;
(2)当a
n=n·f(n), n∈N
*,求证a
1+a
2+…+a
n<2;
(3)设bn=
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325113005-27929.gif)
.
答案
【答案】(1)f(n)}为首页f(n)=
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325113005-17075.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325113005-45873.gif)
(2)略
(3)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325113005-81589.gif)
=
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325113006-78078.gif)
解析
【解析】(1)解:令x=n,y=1得,f(n+1)=f(n)·f(1),
∴{f(n)}为首页f(n)=
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325113005-17075.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325113005-45873.gif)
.
(2)证明a
n=n·f(n),= n·(
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325113005-51931.gif)
)
n,由错位相减法证得a
1+a
2+a
3+…+a
n=2-
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325113006-42125.gif)
;
(3)解:∵
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325113006-64704.gif)
,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325113006-78407.gif)
.
故
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325113007-71450.gif)
.
∴
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325113005-81589.gif)
=
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325113007-36444.gif)
=
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325113006-78078.gif)
举一反三
【题文】已知函数f(x)满足f(x+y)=f(x)·f(y),且f(1)=
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325112948-68206.gif)
.
(1)当n∈N
*时,求f(n)的表达式;
(2)当a
n=n·f(n), n∈N
*,求证a
1+a
2+…+a
n<2;
(3)设bn=
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325112949-13088.gif)
.
【题文】已知函数
f(
x)=
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325112857-15379.gif)
若
f(
f(0))=4
a,则实数
a=
.
【题文】函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325112855-63072.gif)
( )
A.图象无对称轴,且在R上不单调 |
B.图象无对称轴,且在R上单调递增 |
C.图象有对称轴,且在对称轴右侧不单调 |
D.图象有对称轴,且在对称轴右侧单调递增 |
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