Question

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

Answer 1

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