1. **State the problem:** We have an arithmetic sequence with terms $2k + 2$, $5k + 3$, ..., $518$ and the sum of all terms is $5628$. We need to find the value of $k$. 2. **Identify the first term ($a$) and the common difference ($d$):** The first term is $a = 2k + 2$. The second term is $5k + 3$. The common difference is $d = (5k + 3) - (2k + 2) = 3k + 1$. 3. **Find the number of terms ($n$):** The last term $l = 518$. Using the formula for the $n$-th term of an arithmetic sequence: $$l = a + (n - 1)d$$ Substitute values: $$518 = (2k + 2) + (n - 1)(3k + 1)$$ Rearranged: $$518 - 2k - 2 = (n - 1)(3k + 1)$$ $$516 - 2k = (n - 1)(3k + 1)$$ 4. **Use the sum formula for arithmetic series:** $$S_n = \frac{n}{2} (a + l) = 5628$$ Substitute $a$ and $l$: $$5628 = \frac{n}{2} ((2k + 2) + 518) = \frac{n}{2} (2k + 520)$$ Multiply both sides by 2: $$11256 = n(2k + 520)$$ 5. **Express $n$ from the sum equation:** $$n = \frac{11256}{2k + 520}$$ 6. **Express $n$ from the last term equation:** From step 3: $$n - 1 = \frac{516 - 2k}{3k + 1}$$ So: $$n = 1 + \frac{516 - 2k}{3k + 1} = \frac{3k + 1 + 516 - 2k}{3k + 1} = \frac{k + 517}{3k + 1}$$ 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 the quadratic equation:** $$33768k + 11256 = 2k^2 + 1554k + 268840$$ Bring all terms to one side: $$0 = 2k^2 + 1554k + 268840 - 33768k - 11256$$ Simplify: $$0 = 2k^2 - 32214k + 257584$$ 10. **Divide entire equation by 2 to simplify:** $$0 = k^2 - 16107k + 128792$$ 11. **Solve quadratic equation:** Use quadratic formula: $$k = \frac{16107 \pm \sqrt{16107^2 - 4 \times 1 \times 128792}}{2}$$ Calculate discriminant: $$\Delta = 16107^2 - 4 \times 128792 = 259429249 - 515168 = 258914081$$ $$\sqrt{\Delta} \approx 16090.4$$ 12. **Calculate roots:** $$k_1 = \frac{16107 + 16090.4}{2} = \frac{32197.4}{2} = 16098.7$$ $$k_2 = \frac{16107 - 16090.4}{2} = \frac{16.6}{2} = 8.3$$ 13. **Check for integer or reasonable value:** Since $k$ is likely an integer, $k = 8.3$ is not integer, but close to 8. Let's check $k=8$. 14. **Verify $k=8$:** Calculate $n$ from step 6: $$n = \frac{8 + 517}{3 \times 8 + 1} = \frac{525}{25} = 21$$ Calculate sum: $$S_n = \frac{21}{2} (2 \times 8 + 2 + 518) = \frac{21}{2} (16 + 2 + 518) = \frac{21}{2} (536) = 21 \times 268 = 5628$$ Sum matches perfectly. **Final answer:** $$k = 8$$