The sum of the first \( n \) terms of an arithmetic sequence is given by \( S_n = \frac{n}{2}(2a + (n-1)d) \). If the first term is 5 and the common difference is 3, find \( S_{15} \).
["Understanding the Sum of an Arithmetic Sequence: A Step-by-Step Calculation", "Are you looking to master the formula for the sum of the first ( n ) terms of an arithmetic sequence? Understanding this concept not only boosts your mathematical fluency but also simplifies solving real-world problems involving patterns and cumulative growth. In this article, we’ll explore the standard formula and apply it to a concrete example—finding the sum of the first 15 terms when the first term is 5 and the common difference is 3.", "---", "### What is the Sum Formula for an Arithmetic Sequence?", "The sum of the first ( n ) terms of an arithmetic sequence is given by:", "[\nS_n = \frac{n}{2} \left( 2a + (n - 1)d \right)\n]", "where:\n- ( S_n ) = sum of the first ( n ) terms\n- ( a ) = first term\n- ( d ) = common difference\n- ( n ) = number of terms", "This formula offers a fast and efficient way to calculate total sums without manually adding each term—ideal for sequences with consistent differences, like savings growth, evenly spaced heights, or batching production outputs.", "---", "### Applying the Formula: Step-by-Step Example", "Let’s solve a practical case using the given values:\n- First term ( a = 5 )\n- Common difference ( d = 3 )\n- Number of terms ( n = 15 )", "Plug these into the formula:", "[\nS_{15} = \frac{15}{2} \left( 2(5) + (15 - 1)(3) \right)\n]", "Simplify inside the parentheses:", "[\n2(5) = 10\n]\n[\n15 - 1 = 14,\quad 14 \ imes 3 = 42\n]\n[\n10 + 42 = 52\n]", "Now compute the sum:", "[\nS_{15} = \frac{15}{2} \ imes 52 = 15 \ imes 26 = 390\n]", "---", "### Conclusion: The Sum of the First 15 Terms is 390", "Using the arithmetic series formula, we found that the sum ( S_{15} ) of the first 15 terms, where the first term is 5 and the common difference is 3, equals 390.", "Mastering this formula empowers you to quickly analyze patterns in data, forecast cumulative values, and solve complex problems with confidence. Whether in finance, engineering, or everyday calculations, understanding how to compute the sum of an arithmetic sequence is an essential tool for logical reasoning and precision.", "If you want to deepen your math skills, practice multiple problems and explore variations of arithmetic and geometric sequences—your path to mastery starts with understanding the sum formula!", "---", "Keywords: arithmetic sequence sum formula, sum of first n terms, arithmetic series, mathematical formula explanation, step-by-step arithmetic calibration, 'Sₙ = n/2 (2a + (n−1)d)', find S₁₅, 5, 3, 15, cumulative sum calculation"]
