Question

用数学归纳法证明:x^2n-1+y^2n-1能被x+y整除

Answer 1

1.n=1,显然能整除

2.假设n=k（k>1)能被整除,即x^(2k-1)+y^(2k-1)=A(x+y)

当n=k+1时,x^(2k+1)+y^(2k+1)

=x^(2k-1)*x^2+y^(2k-1)*y^2

=A(x+y)-y^(2k-1)*x^2+y^(2k-1)*y^2

=A(x+y)+y^(2k-1)(y^2-x^2)

=A(x+y)+y^(2k-1)(y+x)(y-x)

以上两项都能被（x+y)整除,得证