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

["Understanding the Sum of an Arithmetic Series: A Step-by-Step Guide with Example", "The formula for the sum of the first ( n ) terms of an arithmetic series is given by:\n[\nS_n = \frac{n}{2} \left(2a + (n - 1)d \right)\n]\nwhere:\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.", "This formula is essential for quickly calculating series sums without adding each term individually. When specific values are known, plugging them into the formula becomes fast and accurate.", "---", "### Applying the Formula with Given Values", "In this example, we are given:\n- ( a = 5 ) (first term),\n- ( d = 3 ) (common difference),\n- ( n = 10 ) (number of terms).", "Substitute these values into the arithmetic series sum formula:\n[\nS_{10} = \frac{10}{2} \left(2 \cdot 5 + (10 - 1) \cdot 3 \right)\n]", "Simplify inside the parentheses:\n[\n= 5 \left(10 + 9 \cdot 3 \right)\n]\n[\n= 5 \left(10 + 27 \right)\n]\n[\n= 5 \cdot 37\n]\n[\n= 185\n]", "---", "### Final Answer: Sum of the First 10 Terms", "Thus, the sum of the first 10 terms of the arithmetic series with first term 5 and common difference 3 is:\n[\n\boxed{185}\n]", "Using the formula avoids lengthy calculations and ensures precision—perfect for quick problem-solving in math competitions, classroom exercises, or real-life applications involving linear growth patterns."]