Let's go step by step.
STEP 1 : The amount of liquid left in the mixture is $ x - y = x \cdot \left( 1 - \frac{y}{x} \right) $.
Intuition step : Since $ y $ units of water is replaced in every step, so total amount of mixture is always $x$ units. The ratio of the liquid to water is $ \left( 1 - \frac{y}{x} \right) $
STEP 2 : Again we take out $ y $ units. The amount of liquid left is $ (x - y) \times \texttt{ratio} = x \cdot \left( 1 - \frac{y}{x} \right)^2 $
STEP n : We get your result. $ x \cdot \left( 1 - \frac{y}{x} \right)^n $
And as most mathematics books say, you can prove this by *chuckles* induction.
-sudoankit
