Answer
See below.
Work Step by Step
Proofs using mathematical induction consist of two steps:
1) The base case: here we prove that the statement holds for the first natural number.
2) The inductive step: assume the statement is true for some arbitrary natural number $n$ larger than the first natural number. Then we prove that then the statement also holds for $n + 1$.
Hence, here we have:
1) For $n=1: x^{2(1)-1}-y^{2(1)-1}=x-y$, $x-y$ is a factor of this.
2) Assume for $n=k: a_k=x^{2k-1}-y^{2k-1}$ has $x-y$ as one of its factors. Then for $n=k+1$:
$a_{k+1}=x^{2(k+1)-1}-y^{2(k+1)-1}=x^{2k-1}(x^2-y^2)+(x^{2k-1}-y^{2k-1})y^2=x^{2k-1}(x+y)(x-y)+(x^{2k-1}-y^{2k-1})y^2$. $x-y$
which is a factor of the first term by the inductive hypothesis and it is obviously a factor of the second term. Thus, it is a factor of their sum as well.
Thus we proved what we wanted to.