Let the given statement be P(n), i.e., P (n) : $x^{2n}-y^{2n}$ is divisible by x + y. For n = 1, P(1) : $x^2 – y^2$ is divisible by x + y. Hence, P(1) is true. Assume P(k) is true for some positive integer k, i.e., $x^{2k}-y^{2k}$ = (x + y) g , where g ∊ N ... (i) Now we shall prove that P(k + 1) is true. For this we have to prove that $x^{2(k+1) }-y^{2(k+1)}$ is divisible by x + y. Let us consider, $x^{2(k+1) }-y^{2(k+1)}$ = $x^{2k} x^2 - y^{2k} y^2$ = [(x + y)g + $y^2k$] $x^2 – y^2ky^2$ [From (i)] = (x + y)g $x^2$ + $x^2y^{2k}$ – $y^{2k}y^2$ = (x + y)g $x^2$ + $y^{2k}(x^2 – y^2)$ = (x + y) [$gx^2 + y^{2k}$(x – y)] ⇒ $x^2$ (k + 1) – $y^2$ (k + 1) is divisible by x + y. Hence, P(k + 1) is true, whenever P(k) is true. Hence, by the principle of mathematical induction P(n) is true ∀ n ∈N.