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Q.
For all $n \in N, x^{2 n}-y^{2 n}$ is divisible by
Principle of Mathematical Induction
Solution:
Let the statement $P(n)$ be defined as
$P(n): x^{2 n}-y^{2 n}$ is divisible by $x+y$.
Step I For $n=1, P(1): x^2-y^2=(x-y)(x+y)$
which is divisible by $(x+y)$, that is true.
Step II Let it is true for $n=k$,
i.e., $ x^{2 k}-y^{2 k}=\lambda(x+y) ....$(i)
Step III For $n=k+1$,
$x^{2(k+1)} -y^{2(k+1)}=x^{2 k+2}-y^{2 k+2} $
$ =x^{2 k} x^2-y^{2 k} y^2$
$ =\left[\lambda(x+y)+y^{2 k}\right] x^2-y^{2 k} y^2 \text { [using Eq. (i)] } $
$=\lambda(x+y) x^2+y^{2 k} x^2-y^{2 k} y^2 $
$ =\lambda(x+y) x^2+y^{2 k}\left(x^2-y^2\right) $
$ =\lambda(x+y) x^2+y^{2 k}(x-y)(x+y) $
$ =(x+y)\left[\lambda x^2+y^{2 k}(x-y)\right]$
which is a multiple of $(x+y)$ i.e., divisible by $(x+y)$.
Therefore, $P(k+1)$ is true when $P(k)$ is true. Hence, from the principle of mathematical induction, the statement is true for all natural numbers $n$.