用数学归纳法证明证明x^2n-y^2n能被x+y整除

1.当n=1时
原式=x^2-y^2=(x-y)(x+y)
能被x+y整除
故命题成立
2.假设n=k时命题成立,即 x^(2k)-y^(2k)能被x+y整除
当n=k+1时
x^(2k+2)-y^(2k+2)
=x·x^(2k+1)-y·y^(2k+1)
=(x+y)[x^(2k+1)-y^(2k+1)]-y·x^(2k+1)+x·y^(2k+1)
=(x+y)[x^(2k+1)-y^(2k+1)]-xy[x^(2k)-y^(2k)]
所以,当n=k+1时,命题成立
综上1、2可知
命题成立

n=1时
x^2n-y^2n=x^2-y^2=(x+y)(x-y)
能被X+Y整除
设n≤k时,x^2n-y^2n,能被X+Y整除
n=k+1时
x^2n-y^2n=x^(2k+2)-y^(2k+2)
=(x^2k-y^2k)(x^2+y^2)-x^2ky^2+x^2y^2k
=(x^2k-y^2k)(x^2+y^2)-x^2y^2(x^(2...