Let p and n be real. Now show that

$\displaystyle

n^p < \frac{(n+1)^{p+1} - n^{p+1}}{p+1} < (n+1)^p \\

$

using the identity:

$\displaystyle

b^n - a^n = (b-a)(b^{n-1} + b^{n-2}a + \dotsb + ba^{n-2} + a^{n-1}), n \in \mathbb{N}

$

I have been trying at this one for hours, it is driving me nuts. Can someone point me in the right direction ?