Question

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

Answer 1

Let the given statement be $P\left(n\right)$ i.e.
$P\left(n\right):x^{2n}-y^{2n}$ is divisible by $x+y$.
$P\left(n\right)$ is true for $n=1$ since $x^{2\times 1}-y^{2\times 1}=(x+y)(x-y)$ is divisible by $(x+y)$.
Let $P\left(k\right)$ be true for some positive integer $k$, i.e.
$x^{2k}-y^{2k}$ is divisble by $x+y$
$x^{2k}-y^{2k}=m(x+y)$, where $m\in N......\left(i\right)$
We shall now prove that $P(k+1)$ is true whenever $P\left(k\right)$ is true.
Consider
$x^{2(k+1)}-y^{2(k+1)}$
$=x^{2k}.x^2-y^{2k}.y^2$
$=x^2\left\{m\right(x+y)+y^{2k}\}-y^{2k}.y^2$
$=m(x+y)x^2+y^{2k}(x^2-y^2)$
$=m(x+y)x^2+y^{2k}(x+y)(x-y)$
$=(x+y)\{m{x}^2+y^{2k}(x-y\left)\right\}$
Thus, $P(k+1)$ is true whenever $P\left(k\right)$ is true.
Hence, by the principle of mathematical induction, statement $P\left(n\right)$ is true for all natural numbers i.e., $n$.