**Question:**

Using the principle of mathematical induction, prove each of the following for all $n \in N:$

**Solution:**

To Prove:

$x^{2 n}-y^{2 n}$ is divisible by $x+y$

Let us prove this question by principle of mathematical induction (PMI) for all natural numbers

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

For $\mathrm{n}=1 \mathrm{P}(\mathrm{n})$ is true since $^{x^{2 n}}-y^{2 n}=x^{2}-y^{2}=(x+y) \times(x-y)$

Which is divisible by x + y

Assume P(k) is true for some positive integer k , ie,

$=x^{2 k}-y^{2 k}$ is divisible by $x+y$

Let $x^{2 k}-y^{2 k}=m \times(x+y)$, where $m \in \mathbf{N} \ldots(1)$

We will now prove that P(k + 1) is true whenever P( k ) is true

Consider,

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

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

$=x^{2}\left(x^{2 k}-y^{2 k}+y^{2 k}\right)-y^{2 k} \times y^{2}$ [Adding and subtracting $y^{2 k}$ ]

$=x^{2}\left(\mathrm{~m} \times(\mathrm{x}+\mathrm{y})+y^{2 k}\right)-y^{2 k} \times y^{2}$ [Using 1]

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

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

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

$=(x+y)\left\{m x^{2}+y^{2 k}(x-y)\right\}$, which is factor of $(\mathrm{x}+\mathrm{y})$

Therefore, P (k + 1) is true whenever P(k) is true

By the principle of mathematical induction, P(n) is true for all natural numbers ie, N

Hence proved