Exercise 3.1.9. (Novosibirsk 2007)
Let $a$ and $b$ be positive numbers, and $n \in N$. Prove that
$(n + 1)(a^{n+1} +b^{n+1}) \ge (a+b)(a^n + a^{n-1}b + · · · +b^n)$
The exercice is taken from that paper on olympiad inequalities. I've been trying to employ the mentioned approaches and theory, but without much success and I'm pretty much clueless as of right now. Any ideas would be greatly appreciated! Thanks in advance!
Hint: $a^{n+1}+b^{n+1}\ge\dfrac{a+b}{2}(a^{n-k}b^{k}+b^{n-k}a^{k})$ for $k\ge0$ by Chebyshev inequality.