已知fx满足f1=1/4,4fx*fy=f(x+y)+f(x-y)(x,y属于R),则f(2010)=?

问题描述:

已知fx满足f1=1/4,4fx*fy=f(x+y)+f(x-y)(x,y属于R),则f(2010)=?

当x=y=0时,4f(0)*f(0)=2f(0) f(0)=1/2
当y=1时,4f(x)*f(1)=f(x+1)+f(x-1)
已知f(1)=1/4
则f(x)=f(x+1)+f(x-1) f(x+1)=f(x)-f(x-1)
所以f(x)=f(x-1)-f(x-2) (1)
f(x-1)=f(x-2)-f(x-3) (2)
(1)+(2)得 f(x)=f(x-3)是周期为3的周期函数
故f(2010)=f(2010-3)=f(2007)=f(2004)=.=f(0)=1/2