Answer
See explanations.
Work Step by Step
Step 1. Prove the statement is true when $n=1$: $x+y$ is a factor of $x^{2-1}+y^{2-1}$ , thus it is true for $n=1$.
Step 2. Assume the statement is true when $n=k$: we have $x+y$ is a factor of $x^{2k-1}+y^{2k-1}$
Step 3. Prove it is true for $n=k+1$: $x^{2k+1}+y^{2k+1}=x^2x^{2k-1}-y^2x^{2k-1}+y^2x^{2k-1}+y^2y^{2k-1}=x^{2k-1}(x^2-y^2)+y^2(x^{2k-1}+y^{2k-1})=x^{2k-1}(x-y)(x+y)+y^2(x^{2k-1}+y^{2k-1})$. As $x+y$ is a factor of both terms, it is also a factor of their sum.
Thus, the statement is also true for $n=k+1$
Step 4. Conclusion: with mathematical induction, we have proved that the statement is true for all natural numbers $n$.
