A Real Analysis textbook says the identity $$b^n-a^n = (b-a)(b^{n-1}+\cdots+a^{n-1})$$ yields the inequality $$b^n-a^n < (b-a)nb^{n-1} \text{ when } 0 < a< b.$$ (Note that $n$ is a positive integer)
No matter how I look at it, the inequality seems to be wrong. Take for instance, the inequality does not hold for $n=1$ when one tries mathematical induction. It does not hold for other values of $n$ too. I guess there is something I am missing here and I will appreciate help.