Help Proving the Average is greater than B^(1/n)

Question

Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Define the following two numbers:

$A = (a_1 + a_2 + \cdots + a_n) /n $ (The average of the numbers)

$B = (a_1 + a_2 + \cdots + a_n)^{1 / n}$

Prove that $A \geq B$.

This is what I'm trying to prove for an assignment, and I'm not sure where to start or how do do it. Yes, I can do a base case, and inductive hypothesis, but I'm not sure where to go from there for proving $k+1$. Any hints are appreciated on how to show that this hold true for all positive integers.

Answer 1

Let $a_1=a_2=a_3=a_4=\frac{1}{16}$. Then $A=\frac{1}{16}$.

Note that $B=(1/4)^{1/4}\gt A$.

Perhaps $B$ is supposed to be $(a_1\cdots a_n)^{1/n}$.

Answer 2

For $a_1=a_2=1/2$ and $n=2$, it is false.