Somme des multiples de 15 (pour éviter le double comptage) : \(15 + 30 + \ldots + 90\).

["Avoid Double Counting: The Somme of All Multiples of 15 from 15 to 90", "When summing multiples of a number—such as (15 + 30 + \ldots + 90)—mathematicians emphasize the importance of preventing double counting to ensure accuracy and efficiency. In this article, we explore how to compute the correct sum of all multiples of 15 from 15 to 90 by correctly identifying the sequence and applying the proper summation method, avoiding redundancy.", "---", "### Understanding the Sequence : Multiples of 15 from 15 to 90", "The numbers (15, 30, 45, 60, 75, 90) are the multiples of 15 within the range. But how do we systematically sum them? The key lies in recognizing that this sequence forms an arithmetic series:", "- First term ((a)): 15\n- Common difference ((d)): 15\n- Last term ((l)): 90", "To compute the sum, begin by identifying how many terms ((n)) exist in the sequence. Using:\n[\nn = \frac{l - a}{d} + 1 = \frac{90 - 15}{15} + 1 = \frac{75}{15} + 1 = 5 + 1 = 6\n]", "So, there are 6 terms.", "This directly applies the formula for the sum (S_n) of the first (n) terms of an arithmetic sequence:\n[\nS_n = \frac{n}{2} \ imes (a + l)\n]", "Substitute the known values:\n[\nS_6 = \frac{6}{2} \ imes (15 + 90) = 3 \ imes 105 = 315\n]", "---", "### Why Avoid Double Counting?", "Double counting occurs when a term is included more than once—either by misidentifying sequence terms or miscalculating groupings. In summing multiples of 15, this could happen if one mistakenly adds a term in multiple parts or lists overlapping values.", "For example, incorrectly treating (30) as appearing twice (e.g., as both second and third term) would inflate the sum. Careful alignment with arithmetic progression principles ensures each term is counted exactly once.", "---", "### Final Results and Insights", "- Sum: (15 + 30 + 45 + 60 + 75 + 90 = 315)\n- Confirmed via arithmetic series formula\n- Key principle: Identify the number of terms exactly and avoid redundant counting", "Understanding and applying these rules prevents errors and provides clarity in summation problems involving multiples. Whether studying arithmetic sequences or preparing for mathematical competitions, mastering this method enhances both accuracy and confidence.", "---", "In summary, the correct sum of all multiples of 15 from 15 to 90 is (315), derived cleanly by recognizing the sequence as arithmetic and applying the sum formula precisely—ensuring no double counting and no room for error."]