Sam writes down the numbers 1, 3, 8. Max then adds the same number x to each of Sam’s numbers, to obtain a geometric sequence. What is x?

So we know that $\dfrac{3+x}{1+x} = k$, and $\dfrac{8+x}{3+x} = k$

8 + x = 3k + kx

3 + x = k + kx

Subtracting the second equation from the first one, we get 5 = 2k 

This means k = 5/2 = 2.5 :D

Putting that into the second equation, we get:

3 + x = 2.5 + 2.5x

0.5 = 1.5x

x = 0.5 / 1.5 = 1/3

So, your answer is $\fbox{$\dfrac{1}{3}$}$ :D