I'm trying to figure out why $b^n - a^n < (b - a)nb^{n-1}$.
Using just algebra, we can calculate
$ (b - a)(b^{n-1} + b^{n-2}a + \ldots + ba^{n-2} + a^{n-1}) $
$ = (b^n + b^{n-1}a + \ldots + b^{2}a^{n-2} + ba^{n-1}) - (b^{n-1}a + b^{n-2}a^2 + \ldots + ba^{n-1} + a^{n-1}) $
$ = b^n - a^n, $
but why is it necessarily true that $(b - a)(b^{n-1} + b^{n-2}a + \ldots + ba^{n-2} + a^{n-1}) < (b - a)nb^{n-1}$?
Note: I am interested in an answer to that last question, rather than in another way to prove the general inequality in the title...