Prove the following using the principle of mathematical induction for all n ∈ N

x²ⁿ – y²ⁿ is divisible by x + y.

Let the given statement be P(n), as

P(n): x²ⁿ – y²ⁿ is divisible by (x + y).

First, we check if it is true for n = 1,

P(1): x² - y² = (x - y)(x + y);

∴ It is true for n = 1.

Now we assume that it is true for some positive integer k, such that

P(k):x^2k - y^2k = m(x + y) where m ∈ N.

x^2k = y^2k + m(x + y) ………….(1)

We shall prove that P(k + 1) is true,

P(k + 1):x^{2k + 2} - y^{2k + 2}

⇒ x^2k.x² - y^{2k + 2}

⇒ [y^2k + m(x + y)]x² - y^{2k + 2} From equation(1)

⇒ m(x + y)x² + y^2k(x² - y²)

⇒ m(x + y)x² + y^2k(x - y)(x + y)

⇒ (x + y)[mx² + y^2k(x - y)]

We proved that P(k + 1) is true.

Hence by principle of mathematical induction it is true for all n ∈ N.