a+b+c=0 a^3+b^3+c^3=0 证明：对任意正奇数n,有a^n+b^n+c^n=0

a+b+c=0 a^3+b^3+c^3=0 证明：对任意正奇数n,有a^n+b^n+c^n=0
a+b+c=0 a^3+b^3+c^3=0 证明：对任意正奇数n,有a^n+b^n+c^n=0

a+b+c=0 a^3+b^3+c^3=0 证明：对任意正奇数n,有a^n+b^n+c^n=0
c=-a-b
c^3=-a^3-3a^2b-3ab^2-b^3
所以a^3+b^3-a^3-3a^2b-3ab^2-b^3=0
3ab(a+b)=0
a=0或b=0或a+b=0
若a=0,则c=-b
n是奇数,(-b)^n=-b^n,a^n+b^n+c^n=0+b^n-b^n=0
同理b=0也一样
若a+b=0,则c=0,b=-a,a^n+b^n+c^n=a^n-a^n+0=0
综上
对任意正奇数n,有a^n+b^n+c^n=0