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),求f(2010)
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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),求f(2010)值.
4 f (0) f (0) = 2 f (0)
f (0) = 1/2
4 f (0) f (x) = f (x) + f (x) = 2 f (x)
4 f (x) f (1) = f (x + 1) + f (x - 1)
f (x + 1) = f (x) - f (x - 1)
f (x) = f (x - 1) - f (x - 2)
事实上可以求得通项公式
(1/4)*(1/2+(1/2*I)*sqrt(3))^n+(1/4)*(1/2-(1/2*I)*sqrt(3))^n
解得
f(2010) = 1/2
f (0) = 1/2
4 f (0) f (x) = f (x) + f (x) = 2 f (x)
4 f (x) f (1) = f (x + 1) + f (x - 1)
f (x + 1) = f (x) - f (x - 1)
f (x) = f (x - 1) - f (x - 2)
事实上可以求得通项公式
(1/4)*(1/2+(1/2*I)*sqrt(3))^n+(1/4)*(1/2-(1/2*I)*sqrt(3))^n
解得
f(2010) = 1/2
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