Question

用数学归纳法证明,当n为正奇数时,x^n+y^n能被x+y整除

Answer 1

当n=1时 x+y能被x+y整除

当n=3时 x^3+y^3=(x+y)(x^2-xy+y^2)能被x+y整除

假设当n=2k-1时 x^(2k-1)+y^(2k-1)能被x+y整除

和当n=2k+1时 x^(2k+1)+y^(2k+1)能被x+y整除

当n=2k+3时

x^(2k+3)+y^(2k+3)

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

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

由归纳假设[x^(2k+1)+y^(2k+1)](x^2+y^2)和[x^(2k-1)+y^(2k-1)]x^2*y^2能被x+y整除

所以x^(2k+3)+y^(2k+3)能被x+y整除