<s>[INST] The inequality $b^n - a^n < (b - a)nb^{n-1}$ 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...
 [/INST] You ask why it is necessarily true that 
$$(b - a)(b^{n-1} + b^{n-2}a + \ldots + ba^{n-2} + a^{n-1}) < (b - a)nb^{n-1}.$$  A quick answer is that it is not necessarily true.  We cannot have strict inequality if $b=a$. And there are other issues. For example, if $n=1$, we always have equality. And the inequality sometimes reverses when one of $a$ or $b$ is negative.
We deal in detail with the cases where $a$ and $b$ are both $\ge 0$.  Suppose first that $b>a$, and let $n>1$. Then we want to show that 
$$b^{n-1} + b^{n-2}a + \ldots + ba^{n-2} + a^{n-1} < nb^{n-1}.$$ 
The term $b^{n-2}a$ is strictly less than $b^{n-1}$, and there is such a term, since $n>1$. The remaining terms (if any) are also strictly less than $b^{n-1}$. There is a total of $n$ terms, so their sum is strictly less than $nb^{n-1}$.
A similar argument deals with $b<a$.  Recall that inequality is reversed when we divide both sides by a negative number. So we want to show that 
$$b^{n-1} + b^{n-2}a + \ldots + ba^{n-2} + a^{n-1} > nb^{n-1}.$$
Since $b<a$, the term $a^{n-1}$ is strictly bigger than $b^{n-1}$. We don't even need $n>1$. All the other terms are $\ge b^{n-1}$. There is a total of $n$ terms, and the inequality follows.
We stop at this point. The work is incomplete, since we have not dealt with negative numbers. Let $n=3$, $b=0$, and $a=-1$.  Then $(b-a)(b^2+ba+a^2)$ is positive, but $(b-a)(3)b^2$ is $0$, so the inequality does not hold. With some work, one can track down all the situations where the inequality does hold.  
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