Prove by using the principle of mathematical induction that for all $n \in N$,$x^{2n}-y^{2n}$ is divisible by $x+y$.

(N/A) Let the given statement be $P(n)$,i.e.,$P(n): x^{2n}-y^{2n}$ is divisible by $x+y$.
Step $1$: Check for $n=1$.
$P(1): x^{2(1)}-y^{2(1)} = x^2-y^2 = (x+y)(x-y)$,which is clearly divisible by $(x+y)$. Thus,$P(1)$ is true.
Step $2$: Assume $P(k)$ is true for some positive integer $k$,i.e.,$x^{2k}-y^{2k} = m(x+y)$ for some integer $m$. .......$(i)$
Step $3$: Prove $P(k+1)$ is true.
Consider $x^{2(k+1)}-y^{2(k+1)} = x^{2k} \cdot x^2 - y^{2k} \cdot y^2$.
$= x^2(x^{2k} - y^{2k} + y^{2k}) - y^{2k} \cdot y^2$
$= x^2(x^{2k} - y^{2k}) + y^{2k}(x^2 - y^2)$
$= x^2[m(x+y)] + y^{2k}(x+y)(x-y)$ [Using $(i)$]
$= (x+y)[m x^2 + y^{2k}(x-y)]$.
Since $(x+y)$ is a factor,$P(k+1)$ is true whenever $P(k)$ is true.
Hence,by the principle of mathematical induction,$P(n)$ is true for all $n \in N$.