sequences and series - $A_n$=$frac {a_1+a_2+...a_n}{n}$ is monotonic if $a_n$ is monotonic

if $a_n$ is monotonic increasing/decreasing show that sequence
$A_n$=$\frac {a_1+a_2+...a_n}{n}$ is also monotonic increasing/decreasing.

my attempt:

I intially thought of using induction since $A_2>A_1$ when $a_2>a_1$ so base case is available. but to prove $A_{n+1}>A_n$ doesnt show up easy. Any other way?

Answer

Proving the generalized case is very similar to the base case, because you can write $A_{n+1} = \frac{n}{n+1}A_n + \frac{1}{n+1} a_{n+1}$, which looks very similar to $A_2 = \frac{1}{2}A_1 + \frac{1}{2} a_{2}$

Basically, it amounts to stating why the relative contribution from $a_{n+1}$ is at least as large as the relative contribution from any of the previous elements $a_j, j1}$).