若x+y=a+b,且x^2+y^2=a^2+b^2,请证明:x^2009+y^2009=a^2009+b^2009

Question

若x+y=a+b,且x^2+y^2=a^2+b^2,请证明:x^2009+y^2009=a^2009+b^2009

Answer 1

x+y=a+b
(x+y)^2=(a+b)^2
x^2+y^2+2xy=a^2+b^2+2ab
而：x^2+y^2=a^2+b^2
所以,xy=ab
设：x^n+y^n=a^n+b^n,在n>0时都成立
则：x^(n+1)+y^(n+1)
=(x^n+y^n)(x+y)-(xy^n+x^ny)
=(x^n+y^n)(x+y)-xy(x^(n-1)+y^(n-1))
=(a^n+b^n)(a+b)-ab(a^(n-1)+b^(n-1))
=a^(n+1)+b^(n+1)
n=2008时
x^2009+y^2009=a^2009+b^2009