A Power Mean Versus an Average

★ 8/10Substitution algebra Binomial Theorem inequality

Problem

Let $a > b > 0$ and $n \in \mathbb{N}^*$ . Define

A_n = \frac{a^n + a^{n-1}b + \cdots + ab^{n-1} + b^n}{n+1}, \qquad B_n = \left(\frac{a+b}{2}\right)^n.

1. Prove $A_2 > B_2$ . 2. Compare $A_n$ and $B_n$ in general, with proof.

Answer

Solution

Difficulty	8/10
Topics	Substitution, algebra, Binomial Theorem, inequality

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