1. **State the problem:** We have an arithmetic sequence with terms $(2k+2), (5k+3), \ldots, 518$ and the sum of all terms is 5628. We need to find the value of $k$. 2. **Identify the first term $a$ and common difference $d$:** - First term $a = 2k + 2$ - Second term $= 5k + 3$ - Common difference $d = (5k + 3) - (2k + 2) = 3k + 1$ 3. **Find the number of terms $n$:** The last term $l = 518$ is given by the $n$-th term formula: $$l = a + (n-1)d$$ Substitute: $$518 = (2k + 2) + (n-1)(3k + 1)$$ 4. **Sum of arithmetic series formula:** $$S_n = \frac{n}{2} (a + l)$$ Given $S_n = 5628$, so: $$5628 = \frac{n}{2} \big((2k + 2) + 518\big) = \frac{n}{2} (2k + 520)$$ 5. **Express $n$ from the sum equation:** $$5628 = \frac{n}{2} (2k + 520) \implies n = \frac{2 \times 5628}{2k + 520} = \frac{11256}{2k + 520}$$ 6. **Express $n$ from the last term equation:** From step 3: $$518 = 2k + 2 + (n-1)(3k + 1)$$ Rearranged: $$(n-1)(3k + 1) = 518 - 2k - 2 = 516 - 2k$$ $$n - 1 = \frac{516 - 2k}{3k + 1}$$ $$n = 1 + \frac{516 - 2k}{3k + 1} = \frac{3k + 1 + 516 - 2k}{3k + 1} = \frac{k + 517}{3k + 1}$$ 7. **Set the two expressions for $n$ equal:** $$\frac{11256}{2k + 520} = \frac{k + 517}{3k + 1}$$ Cross multiply: $$11256(3k + 1) = (k + 517)(2k + 520)$$ 8. **Expand both sides:** Left: $$11256 \times 3k + 11256 = 33768k + 11256$$ Right: $$(k)(2k) + k(520) + 517(2k) + 517(520) = 2k^2 + 520k + 1034k + 268840 = 2k^2 + 1554k + 268840$$ 9. **Form quadratic equation:** $$33768k + 11256 = 2k^2 + 1554k + 268840$$ Bring all terms to one side: $$0 = 2k^2 + 1554k + 268840 - 33768k - 11256$$ $$0 = 2k^2 - 32214k + 257584$$ 10. **Simplify by dividing all terms by 2:** $$0 = k^2 - 16107k + 128792$$ 11. **Solve quadratic equation:** Use quadratic formula: $$k = \frac{16107 \pm \sqrt{16107^2 - 4 \times 1 \times 128792}}{2}$$ Calculate discriminant: $$16107^2 = 259429249$$ $$4 \times 128792 = 515168$$ $$\sqrt{259429249 - 515168} = \sqrt{258914081} \approx 16087.9$$ 12. **Calculate roots:** $$k_1 = \frac{16107 + 16087.9}{2} = \frac{32194.9}{2} = 16097.45$$ $$k_2 = \frac{16107 - 16087.9}{2} = \frac{19.1}{2} = 9.55$$ 13. **Check for valid $k$:** Since $k$ is likely an integer or a reasonable number, $k \approx 9.55$ is plausible. **Final answer:** $$k \approx 9.55$$