new arithmetic mean

The total of the first six numbers is $6(20.5)=123$ . Let $x$ be the larger number that was added and $y$ be the smaller number. Then we have

$\dfrac{123+x+y}{8}=17.5$ $\implies$ $x+y=19$ $\color{#D61F06}(1)$

Another equation is

$x-y=3$ $\color{#D61F06}(2)$

Adding the two equations, we obtain

$2x=22$

$x=11$

Finally, the other number is

$11-y=3$

$11-3=y$

$y=8$

So the numbers are $8$ and $11$ .