1. We are given a geometric progression with terms $2$, $x$, $y$, and $250$. 2. In a geometric progression, the ratio between consecutive terms is constant. Let this common ratio be $r$. 3. From the first two terms: $x = 2r$. 4. From the second and third terms: $y = xr = 2r^2$. 5. From the third and fourth terms: $250 = yr = 2r^3$. 6. Solve for $r$ from $250 = 2r^3$: $$r^3 = \frac{250}{2} = 125$$ $$r = \sqrt[3]{125} = 5$$ 7. Compute $x$ and $y$: $$x = 2r = 2 \times 5 = 10$$ $$y = 2r^2 = 2 \times 5^2 = 2 \times 25 = 50$$ 8. Final answer: $x = 10$, $y = 50$