<s>[INST] Prove using mathematical induction that $x^{2n} - y^{2n}$ is divisible by $x+y$ Prove using mathematical induction that 
 $(x^{2n} - y^{2n})$ is divisible by $(x+y)$.
Step 1: Proving that the equation is true for $n=1 $
$(x^{2\cdot 1} - y^{2\cdot 1})$ is divisible by $(x+y)$ 
Step 2: Taking $n=k$
$(x^{2k} - y^{2k})$ is divisible by $(x+y)$
Step 3: proving that the above equation is also true for $(k+1)$
$(x^{2k+2} - y^{2k+2})$ is divisible by $(x+y)$.
Can anyone assist me what would be the next step? Thank You in advance!
 [/INST] When $n=1$,
$$x^{2n} - y^{2n} =x^2-y^2=(x+y)(x-y)$$
which is divisible by $(x+y)$. Assume true for $n=k$. Then $x^{2k}-y^{2k}$ is divisible by $x+y$.
Putting $n=k+1$,
$$x^{2n}-y^{2n}=x^{2k+2}-y^{2k+2}=\left(x^{k+1}-y^{k+1}\right) \left(x^{k+1}+y^{k+1}\right)$$
When $k$ is an odd number, $x+y$ is a factor of $x^{k+1}-y^{k+1}$.
Similarly, when $k$ is an even number, $x+y$ is a factor of $x^{k+1} + y^{k+1}$.
Hence $x^{2n}-y^{2n}$ is divisible by $x+y$ for integers $n>0$.
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