Question

Prove that: $x^{2n}-y^{2n}$ is divisible by x+y.

Answer 1

Let $P\left(n\right):x^{2n}-y^{2n}=(x+y)\times d$ where $dϵN$

For $n=1$

LHS $=x^{2\times 1}-y^{2\times 1}$

$=x^2-y^2$

$(x+y)(x-y)$

=RHS

$\therefore P\left(n\right)$ is true for $n=1$

Assume $P\left(k\right)$ is true.

$x^{2k}-y^{2k}=(x+y)\times m$ where $mϵN$

We will prove that $P(k+1)$ is true.

LHS $=x^{2\times (k+1)}-y^{2\times (k+1)}$

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

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

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

$=(x+y)\times r$

where, $r=[m{x}^2+y^{2k}(x-y\left)\right]$

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

$\therefore$ By the principle of mathematical induction, $P\left(n\right)$ is true for n, where n is a natual number.