Need help with mathematical induction

Question

Suppose $a, b$ are positive real numbers. Then $\frac{a^n + b^n}{2} \ge \left(\frac{a+b}{2}\right)^n$ for any $n$ ∈ $\mathbb{N}$ \ {$0$}.

How should I prove the above statement by mathematical induction? I am stuck in the step in proving $P(k+1)$.

Answer 1

By the induction hypothesis, $$ \left(\frac{a+b}{2}\right)^{\!k+1}=\left(\frac{a+b}{2}\right)^{\!k}\frac{a+b}{2}\le\frac{a^k+b^k}{2}\frac{a+b}{2} $$ Can you compare the last term to $$ \frac{a^{k+1}+b^{k+1}}{2} $$ and finish the task?