The sum of the first \(n\) terms of an arithmetic sequence is given by \(S_n = rac{n}{2} (2a + (n-1)d)\). If the first term \(a = 5\) and the common difference \(d = 3\), find \(S_{10}\).

["# Understanding the Sum of an Arithmetic Sequence: Calculating (S_{10})", "When working with arithmetic sequences, one of the most useful formulas is the sum of the first (n) terms. This formula allows you to calculate the total without needing to add each term individually. For an arithmetic sequence defined by first term (a) and common difference (d), the sum of the first (n) terms is given by:", "[\nS_n = \frac{n}{2} \left(2a + (n-1)d\right)\n]", "This equation provides a fast and efficient way to compute cumulative sums—critical in both academic and practical applications.", "## Applying the Formula to a Specific Sequence", "Let’s apply this formula to a concrete example. Suppose in a particular arithmetic sequence:", "- The first term (a = 5)\n- The common difference (d = 3)\n- We want to find the sum of the first 10 terms: (S_{10})", "Substituting these values into the sum formula:", "[\nS_{10} = \frac{10}{2} \left(2 \cdot 5 + (10 - 1) \cdot 3\right)\n]", "Now simplify step by step:", "[\nS_{10} = 5 \left(10 + 9 \cdot 3\right)\n]\n[\nS_{10} = 5 \left(10 + 27\right)\n]\n[\nS_{10} = 5 \cdot 37 = 185\n]", "Thus, the sum of the first 10 terms is 185.", "### Why This Formula Works", "The formula reduces the repetitive process of adding terms into a simple computation. It leverages the symmetry of arithmetic sequences—averaging the first and last terms, then multiplying by the number of terms. This principle allows efficient calculations in fields such as finance, physics, and computer science, where sequential data often follows a linear pattern.", "### Conclusion", "Knowing the sum formula of an arithmetic series—especially with established values of (a) and (d)—is invaluable. In this case, with (a = 5), (d = 3), and (n = 10), we efficiently determined that the sum of the first 10 terms is 185. Mastering such formulas strengthens both mathematical fluency and problem-solving capability in real-world scenarios."]