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 \( a = 3 \) and the common difference \( d = 5 \), find the sum of the first 10 terms.

["Sum of the First ( n ) Terms of an Arithmetic Sequence: A Clear Guide with Example", "Understanding the sum of the first ( n ) terms of an arithmetic sequence is essential for mastering algebra and solving problems in mathematics and real-world applications. The general formula for this sum is:", "[\nS_n = \frac{n}{2} \left( 2a + (n - 1)d \right)\n]", "where:\n- ( S_n ) is the sum of the first ( n ) terms,\n- ( a ) is the first term,\n- ( d ) is the common difference,\n- ( n ) is the number of terms.", "---", "### Applying the Formula to a Real Problem", "Let’s use the given values to compute the sum of the first 10 terms of an arithmetic sequence. Here, the first term ( a = 3 ) and the common difference ( d = 5 ). We want to find ( S_{10} ), the sum of the first 10 terms.", "Step 1: Plug values into the formula", "[\nS_{10} = \frac{10}{2} \left( 2(3) + (10 - 1)(5) \right)\n]", "Step 2: Simplify inside the parentheses", "[\nS_{10} = 5 \left( 6 + 9 \ imes 5 \right) = 5 \left( 6 + 45 \right)\n]", "[\nS_{10} = 5 \ imes 51 = 255\n]", "---", "### Final Answer", "Thus, the sum of the first 10 terms of this arithmetic sequence is 255.", "This formula and method provide a fast and effective way to compute series sums, crucial not only in math competitions but also in fields such as finance, physics, and data analysis. Remember: know ( a ), ( d ), and ( n ), apply the formula confidently, and accuracy follows."]