1. **State the problem:** Given the arithmetic progression (A P) 3, x, y, 18, find the values of $x$ and $y$. 2. **Recall the formula for an arithmetic progression:** The difference between consecutive terms is constant. If $a_1, a_2, a_3, \ldots$ is an A P, then $$a_2 - a_1 = a_3 - a_2 = \cdots = d$$ where $d$ is the common difference. 3. **Apply the formula:** - The first term $a_1 = 3$ - The fourth term $a_4 = 18$ Since the difference is constant, $$x - 3 = y - x = 18 - y$$ 4. **Set up equations:** From $x - 3 = y - x$, we get $$2x = y + 3 \implies y = 2x - 3$$ From $y - x = 18 - y$, we get $$2y = x + 18$$ 5. **Substitute $y$ from the first equation into the second:** $$2(2x - 3) = x + 18$$ $$4x - 6 = x + 18$$ $$4x - x = 18 + 6$$ $$3x = 24$$ $$x = 8$$ 6. **Find $y$:** $$y = 2(8) - 3 = 16 - 3 = 13$$ **Final answer:** $$x = 8, \quad y = 13$$