Simplifying, \( S_{15} = \frac{15}{2} (10 + 42) = \frac{15}{2} \times 52 = 15 \times 26 = 390 \).

["# Simplifying ( S_{15} = \frac{15}{2} (10 + 42) = \frac{15}{2} \ imes 52 = 15 \ imes 26 = 390 ): A Step-by-Step Explanation", "Mathematics often presents complex-looking problems that can be broken down into simpler steps for clarity. One such expression is the sum of the first 15 positive integers, written as ( S_{15} = \frac{15}{2}(10 + 42) ), which simplifies neatly to 390. In this article, we’ll walk through the full derivation in an accessible way to help anyone understand how this elegant mathematical identity works.", "---", "## What Is ( S_{15} )?", "The symbol ( S_{15} ) refers to the sum of the first 15 natural numbers:", "[\nS_{15} = 1 + 2 + 3 + \cdots + 14 + 15\n]", "There is a powerful formula to calculate such sums without adding each number individually—known as the arithmetic series formula:", "[\nS_n = \frac{n}{2}(a_1 + a_n)\n]", "Where:\n- ( n ) is the number of terms,\n- ( a_1 ) is the first term,\n- ( a_n ) is the last term.", "---", "## Applying the Formula to ( S_{15} )", "For ( S_{15} ), identify the values:", "- ( n = 15 ) (since we’re summing 15 terms),\n- ( a_1 = 1 ) (the first positive integer),\n- ( a_n = 15 ) (the last term).", "Substitute into the formula:", "[\nS_{15} = \frac{15}{2}(1 + 15) = \frac{15}{2} \ imes 16\n]", "Wait—here’s where a common twist appears: the expression given in the prompt uses ( \frac{15}{2}(10 + 42) ), which might seem irregular. Let’s explore why this form appears and how it connects.", "---", "## Understanding the Alternative Interpretation ( \frac{15}{2}(10 + 42) )", "The form ( \frac{15}{2}(10 + 42) ) simplifies mathematically as follows:", "[\n10 + 42 = 52 \quad \Rightarrow \quad S_{15} = \frac{15}{2} \ imes 52\n]", "Since ( \frac{15}{2} \ imes 52 = 15 \ imes 26 = 390 ), this confirms the sum:", "[\n\boxed{S_{15} = 390}\n]", "But why does the sum appear as ( \frac{15}{2}(10 + 42) )? This form reflects how the first term and the last term are encoded in a flexible way—sometimes expressed relative to other reference points. Let's examine this mathematically.", "---", "## How ( 10 + 42 ) Relates to the Sequence", "While the true first term is 1, consider a shift of perspective:\nSuppose you pair terms inward from both ends:", "- ( 1 + 15 = 16 )\n- ( 2 + 14 = 16 )\n- ( 3 + 13 = 16 )\n- ( 4 + 12 = 16 )\n- ( 5 + 11 = 16 )\n- ( 6 + 10 = 16 )\n- Middle term: ( 7 ) (unchonned since ( n = 15 ), odd)", "There are 7 such middle pairs adding to 16, plus one unpaired middle term, 7. So total sum:", "[\nS_{15} = 7 \ imes 16 + 7 = 112 + 7 = 119 \quad \ ext{(Wait—this contradicts earlier result)}\n]", "Ah! There’s a catch: this pairing trick works neatly for sums starting at 1, but the pair sum is not (10 + 42) directly. Instead, consider:", "If we use symmetrical pairing, with ( a_1 = 1 ), ( a_{15} = 15 ), then:", "[\nS_{15} = \frac{15}{2}(1 + 15) = \frac{15}{2} \ imes 16 = 120\n]", "But that’s 120, not 390. So why does ( S_{15} = \frac{15}{2}(10 + 42) )?", "Let’s reverse-engineer this.", "---", "## Deconstructing ( \frac{15}{2}(10 + 42) = \frac{15}{2} \ imes 52 = 390 )", "We know:", "[\nS_{15} = \frac{15}{2} \ imes 16 = 15 \ imes 8 = 120\n]", "Wait—this is confusing. The error lies in how (10 + 42) maps.", "Actually, notice:", "- The average of the first and last term: ( \frac{1 + 15}{2} = 8 )\n- Number of terms: ( n = 15 )\n- Sum: ( 8 \ imes 15 = 120 ) — correct.", "But ( \frac{15}{2}(10 + 42) ):\nCalculate ( 10 + 42 = 52 ), ( \frac{15}{2} \ imes 52 = 15 \ imes 26 = 390 ) — not 120!", "So this expression does not represent ( S_{15} ) directly.", "Hence, ( S_{15} = \frac{15}{2}(1 + 15) = 120 ) is the correct and safest evaluation.", "---", "## Where Does ( \frac{15}{2}(10 + 42) = 390 ) Fit?", "This likely stems from a different sorting or indexing trick, such as shifting or reindexing. For instance:", "Let’s try reordering the sum:", "Suppose you pair smallest and largest, but scale differently — though in this form, the identity holds only as ( \frac{n}{2}(a_1 + a_n) ).", "Alternatively, suppose", "[\nS_{15} = \frac{15}{2}(10 + 42)\n]", "Then unless there's a clever substitution (e.g., weighted terms), this is not standard.", "Conclusion: The cleanest, correct derivation is:", "[\nS_{15} = \frac{15}{2}(1 + 15) = \frac{15}{2} \ imes 16 = 15 \ imes 8 = 120\n]", "But the expression ( \frac{15}{2}(10 + 42) = 390 ) is incorrect as a direct sum, though it equals 390 algebraically:", "[\n\frac{15}{2} \ imes 52 = 390 \quad \ ext{and} \quad 1 + 15 = 16,\ \frac{15}{2} \ imes 16 = 120 <br/>\ne 390\n]", "Thus, either:\n- There’s a typo, or\n- The expression is illustrating a non-canonical rearrangement, perhaps in a geometric or weighted context, but for standard arithmetic, ( S_{15} = 390 ) is not equal to ( \frac{15}{2}(10 + 42) ).", "---", "## Final Clarification and Summary", "While ( S_{15} = \frac{15}{2}(1 + 15) = 120 ) is rigorous, the form ( \frac{15}{2}(10 + 42) = 390 ) appears to be a multiplicative shortcut or rearrangement not standard for this sum.", "Key Takeaway:\n- The arithmetic series formula gives the clearest shortcut:\n [\n S_n = \frac{n}{2}(a + l)\n ]\n where ( a ) is first, ( l ) last, ( n ) count.", "- This yields:\n [\n S_{15} = \frac{15}{2}(1 + 15) = 120\n ]", "- The expression ( \frac{15}{2}(10 + 42) = 390 ) is mathematically valid but not equivalent to ( S_{15} ); it’s either a misinterpretation or a specialized transformation not widely applied.", "---", "## Why This Matters: Clarity Over Confusion", "Math thrives on precision. Recognizing when an expression simplifies cleanly versus when it’s misleading is key. The correct, simplified value of ( S_{15} ) is 120—easily computed in seconds using the series formula.", "Remember:\n[\n\boxed{S_{15} = 120}\n]\nand the form ( S_n = \frac{n}{2}(a + l) ) is your strongest tool.", "---", "## Further Reading", "- Arithmetic Series Formula: How and why ( S_n = \frac{n(a_1 + a_n)}{2} ) works\n- Pairing Techniques in Series Summation\n- Common Tricks and Pitfalls in Mathematical Expression Simplification", "---", "Keywords: ( S_{15} ) simplification, arithmetic series formula, summing integers, ( \frac{n}{2}(a_1 + a_n) ), simplifying sums, step-by-step math, Fibonacci sum misinterpretation, performing algebra correctly."]