Annoying latex \begin{align*} (a^n-b^n)\left(\frac{1}{b^{n-1}}-\frac{1}{a^{n-1}}\right) &= (a^n-b^n)\left(\frac{a^{n-1}-b^{n-1}}{a^{n-1}b^{n-1}}\right) \\ &= (a-b)^2(a^{n-1}+a^{n-2}b+…+b^{n-1})\left(\frac{a^{n-2}+…+b^{n-2}}{a^{n-1}b^{n-1}}\right) \\ &\geq \left(\frac{(\sqrt{a}-\sqrt{b})^2}{(ab)^{n-1}}\right)(\sqrt{a}+\sqrt{b})^2(na^{\frac{n-1}{2}}b^{\frac{n-1}{2}})((n-1)a^{\frac{n-2}{2}}b^{\frac{n-2}{2}}) \\ &\geq n(n-1)\left(\frac{(\sqrt{a}-\sqrt{b})^2}{(ab)^{n-1}}\right)(4\sqrt{ab})((ab)^{\frac{2n-3}{2}}) \\ &= 4n(n-1)(\sqrt{a}-\sqrt{b})^2 \end{align*}Inequality couldn’t holds since $a > b$