1. **State the problem:** We need to find three numbers in geometric progression (GP) whose sum is 26 and the difference between the first and third terms is 18. 2. **Set variables:** Let the three numbers be $a$, $ar$, and $ar^2$, where $a$ is the first term and $r$ is the common ratio. 3. **Write equations from the problem:** - Sum: $$a + ar + ar^2 = 26$$ - Difference: $$ar^2 - a = 18$$ 4. **Simplify the difference equation:** $$a(r^2 - 1) = 18$$ 5. **Express $a$ from the difference equation:** $$a = \frac{18}{r^2 - 1}$$ 6. **Substitute $a$ into the sum equation:** $$\frac{18}{r^2 - 1} + \frac{18r}{r^2 - 1} + \frac{18r^2}{r^2 - 1} = 26$$ 7. **Combine terms:** $$\frac{18(1 + r + r^2)}{r^2 - 1} = 26$$ 8. **Multiply both sides by $r^2 - 1$:** $$18(1 + r + r^2) = 26(r^2 - 1)$$ 9. **Expand right side:** $$18 + 18r + 18r^2 = 26r^2 - 26$$ 10. **Bring all terms to one side:** $$18 + 18r + 18r^2 - 26r^2 + 26 = 0$$ 11. **Simplify:** $$18r + (18r^2 - 26r^2) + (18 + 26) = 0$$ $$18r - 8r^2 + 44 = 0$$ 12. **Rewrite:** $$-8r^2 + 18r + 44 = 0$$ 13. **Multiply both sides by $-1$ to simplify:** $$\cancel{-}8r^2 + 18r + 44 = 0 \Rightarrow 8r^2 - 18r - 44 = 0$$ 14. **Solve quadratic equation:** $$r = \frac{18 \pm \sqrt{(-18)^2 - 4 \times 8 \times (-44)}}{2 \times 8} = \frac{18 \pm \sqrt{324 + 1408}}{16} = \frac{18 \pm \sqrt{1732}}{16}$$ 15. **Calculate $\\sqrt{1732}$:** $$\sqrt{1732} \approx 41.61$$ 16. **Find roots:** $$r_1 = \frac{18 + 41.61}{16} = \frac{59.61}{16} \approx 3.726$$ $$r_2 = \frac{18 - 41.61}{16} = \frac{-23.61}{16} \approx -1.476$$ 17. **Find corresponding $a$ values:** For $r_1 = 3.726$: $$a = \frac{18}{(3.726)^2 - 1} = \frac{18}{13.89 - 1} = \frac{18}{12.89} \approx 1.396$$ For $r_2 = -1.476$: $$a = \frac{18}{(-1.476)^2 - 1} = \frac{18}{2.18 - 1} = \frac{18}{1.18} \approx 15.254$$ 18. **Calculate the three numbers for each case:** - For $r_1$: $$a = 1.396, ar = 1.396 \times 3.726 \approx 5.2, ar^2 = 1.396 \times (3.726)^2 \approx 19.4$$ Sum check: $1.396 + 5.2 + 19.4 = 26$ (approx) - For $r_2$: $$a = 15.254, ar = 15.254 \times (-1.476) \approx -22.5, ar^2 = 15.254 \times (-1.476)^2 \approx 33.3$$ Sum check: $15.254 - 22.5 + 33.3 = 26$ (approx) 19. **Final answer:** The three numbers are approximately either $(1.4, 5.2, 19.4)$ or $(15.3, -22.5, 33.3)$.