Prove by induction that: $$(y-x)x^n \leq \frac{y^{n+1}-x^{n+1}}{n+1} \leq (y-x)y^n\ . $$
As a hint, the professor told us to use the following expression that we had previously proven: $$\sum_{i=0}^n{x^i}= \frac{1-x^{n+1}}{1-x} $$
I already tried several things, but I can’t manage to get to the solution.
I'm not quit sure how to use the hint given by the professor. But here's one way to do this:
Recall that $$(y^{n+1}-x^{n+1}) = (y-x)(y^n+y^{n-1}x+....+yx^{n-1}+x^{n})$$
Now assuming that $x<y$ you can replace all the $y$'s with $x$'s in the second multiple1 and get that $$y^{n+1}-x^{n+1} \geq (y-x) (n+1)x^n$$ Divide by $n+1$ and you get the first inequality. For the other one replace the $x$'s with $y$'s.
1.Edit It's possible that you need to prove this part by induction. That is, prove by induction that $$(n+1)x^n \leq y^n+y^{n-1}x+...+yx^{n-1}+x^n$$