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

Let $P(n)$ be the statement: $x^{2n}-y^{2n}$ is divisible by $x+y$.

Base case: For $n=1$,

which is clearly divisible by $x+y$. So $P(1)$ is true.

Inductive step: Assume $P(k)$ is true for some $k\ge1$, i.e.

We must show $x^{2(k+1)}-y^{2(k+1)}=x^{2k+2}-y^{2k+2}$ is divisible by $x+y$.

Insert and subtract $x^2y^{2k}$ to split the expression using the induction hypothesis:

Now substitute the induction hypothesis $x^{2k}-y^{2k}=(x+y)g(x,y)$ and factor $x^2-y^2=(x-y)(x+y)$: …

Divisibility induction using the standard "add and subtract" trick: split $x^2\cdot x^{2k}-y^2\cdot y^{2k}$ by inserting $\pm x^2y^{2k}$, so one piece becomes $x^2(x^{2k}-y^{2k})$ (which uses the hypothesis) …

• Not inserting the "add and subtract $x^2y^{2k}$" term, which is the key step that lets both pieces be shown divisible by $x+y$ separately.
• Forgetting that $x^2-y^2=(x-y)(x+y)$ is itself divisible by $x+y$ — treating it as an unrelated leftover term instead of factoring it. …