[tex]a^5+b^5 = (a+b)(a^4-a^3b+a^2b^2-ab^3+b^4) \\ \\ (a+b)(a^4-a^3b+a^2b^2-ab^3+b^4) = \\ \\ = a^5-a^4b+a^3b^2-a^2b^3+ab^4+a^4b-a^3b^2+a^2b^3-ab^4+b^5 = \\ \\ = a^5+b^5-a^4b+a^4b+a^3b^2-a^3b^2-a^2b^3+a^2b^3+ab^4-ab^4 = \\ \\ = a^5+b^5\quad (A) \\ \\ \\ a^{n}+b^n = \\ =(a+b)(a^{n-1}-a^{n-2}b+a^{n-3}b^2-....+a^2b^{n-3}-ab^{n-2}+b^{n-1}), \\\\ ,\forall n$ impar.[/tex]
