Question

Using principle of Mathematical induction prove that:
$x^{2n}-y^{2n}$ is divisible by $x+y$, where $n\in N$

Answer 1

Let $P\left(n\right):x^{2n}-y^{2n}$ is divisible by $x+y$

For $n=1,P\left(1\right):x^2-y^2$ is divisible by $x+y$ which is true.

$\therefore P\left(1\right)$ is true.

Let us assume $P\left(k\right)$ is true for some $k\in N$

i.e, $x^{2k}-y^{2k}$ is divisible by $x+y$

Let $x^{2k}-y^{2k}=(x+y)d,$ where $d\in N$

Consider $x^{2(k+1)}-y^{2(k+1)}=x^{2k+2}-y^{2k+2}$

$=x^{2k}{x}^2-y^{2k}{y}^2$

$=x^{2k}{x}^2-x^2{y}^{2k}+x^2{y}^{2k}-y^{2k}{y}^2$

$=x^2(x^{2k}-y^{2k})+y^{2k}(x^2-y^2)$

$=x^2(x+y)d+y^{2k}(x+y)(x-y)$

$=(x+y)[x^{2}d+y^{2k}(x-y\left)\right]$

$\therefore x^{2(k+1)}-y^{2(k+1)}$ is divisible by $x+y$

which is $P(k+1)$

Thus, $P\left(k\right)\Rightarrow P(k+1)$
Hence, by mathematical induction $P\left(n\right)$ is true for all $n\in N$.