已知函数f(x)满足:f(1)=1/4,4f(x)f(y)=f(x+y)+f(x-y)(x,y属于R).则F(2010)=?
希望得到T=6的详细过程,不是只f(n+2)=-f(n-1).
人气:141 ℃ 时间:2019-08-21 20:28:40
解答
取y=0得4f(x)f(0)=2f(x)得f(0)=1/2;
取y=1得 f(x)=f(x+1)+f(x-1)
进一步得 f(x+1)=f(x)+f(x+2)
两式相加得 f(x-1)+f(x+2)=0,
进一步得 f(x+2)+f(x+5)=0,
以上两式相减得f(x-1)=f(x+5),
进一步得 f(x)=f(x+6)
所以T=6;
F(2010)=F(0+335*6)=F(0)=1/2.
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