1. **State the problem:** Solve the equation $$(-4) + (-1) + 2 + 5 + \ldots + x = 437$$ where the terms form an arithmetic sequence. 2. **Identify the sequence:** The terms are $$-4, -1, 2, 5, \ldots$$ 3. **Find the common difference:** $$d = -1 - (-4) = 3$$ 4. **General term formula:** The $n$-th term of an arithmetic sequence is given by $$a_n = a_1 + (n-1)d$$ where $a_1 = -4$ and $d = 3$. 5. **Express $a_n$:** $$a_n = -4 + (n-1) \times 3 = -4 + 3n - 3 = 3n - 7$$ 6. **Sum of first $n$ terms:** The sum $S_n$ is $$S_n = \frac{n}{2} (a_1 + a_n)$$ 7. **Substitute $a_1$ and $a_n$:** $$S_n = \frac{n}{2} (-4 + (3n - 7)) = \frac{n}{2} (3n - 11)$$ 8. **Set sum equal to 437:** $$\frac{n}{2} (3n - 11) = 437$$ 9. **Multiply both sides by 2:** $$n(3n - 11) = 874$$ 10. **Expand:** $$3n^2 - 11n = 874$$ 11. **Bring all terms to one side:** $$3n^2 - 11n - 874 = 0$$ 12. **Solve quadratic equation:** Using the quadratic formula $$n = \frac{11 \pm \sqrt{(-11)^2 - 4 \times 3 \times (-874)}}{2 \times 3} = \frac{11 \pm \sqrt{121 + 10488}}{6} = \frac{11 \pm \sqrt{10609}}{6}$$ 13. **Calculate discriminant:** $$\sqrt{10609} = 103$$ 14. **Find roots:** $$n = \frac{11 + 103}{6} = \frac{114}{6} = 19$$ $$n = \frac{11 - 103}{6} = \frac{-92}{6} = -\frac{46}{3}$$ (discard negative) 15. **Find $x = a_n$ for $n=19$:** $$x = 3(19) - 7 = 57 - 7 = 50$$ **Final answer:** $$x = 50$$