已知函数f(x)对任意的正整数x都有f(x+2)=2f(x+1)-f(x),f(1)=2 f(3)=6,

令f(x)为an
则:
f(x+2)=2f(x+1)-f(x)
等价于:
a(n+2)=2a(n+1)-an
则:
a(n+2)-a(n+1)=a(n+1)-an
[a(n+2)-a(n+1)]/[a(n+1)-an]=1
则:
{a(n+1)-an}为公比为1的等比数列
则:
a(n+1)-an=a2-a1
又a3=2a2-a1
6=2a2-2
则:a2=4
则:
a(n+1)-an=4-2=2
则:{an}为公差为2的等差数列
则:
an=a1+(n-1)d
=2+2(n-1)
=2n
则:f(2008)
=a2008
=2*2008
=4016