Question**: Find the sum of the first 20 terms of the arithmetic sequence where the first term is 5 and the common difference is 3.
["Title: How to Find the Sum of the First 20 Terms in an Arithmetic Sequence | Step-by-Step Guide", "---", "Introduction", "Arithmetic sequences are powerful mathematical tools used in finance, computer science, engineering, and everyday problem-solving. A classic question often asked is: What is the sum of the first 20 terms of an arithmetic sequence where the first term is 5 and the common difference is 3? Answers to such problems involve understanding core concepts and applying a simple formula. In this SEO-optimized guide, we’ll walk you through the steps to calculate the sum efficiently and explain how this method improves learning and search visibility.", "---", "What Is an Arithmetic Sequence?", "An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the previous term. The general form is:", "[ a_n = a_1 + (n - 1)d ]", "Where:\n- ( a_1 ) = first term\n- ( d ) = common difference\n- ( n ) = number of terms", "The sum of the first n terms, denoted ( S_n ), is calculated using the formula:", "[ S_n = \frac{n}{2} \ imes (2a_1 + (n - 1)d) ]", "Or equivalently:", "[ S_n = \frac{n}{2} \ imes (a_1 + a_n) ]", "---", "Given Problem Breakdown", "- First term (( a_1 )) = 5\n- Common difference (( d )) = 3\n- Number of terms (( n )) = 20", "---", "Step-by-Step Calculation", "1. Identify the formula to use\nSince ( a_1 ) and ( d ) are known and ( n = 20 ), we use:\n[\nS_{20} = \frac{20}{2} \ imes (2 \ imes 5 + (20 - 1) \ imes 3)\n]", "2. Simplify inside the parentheses\nCalculate step-by-step:\n[\n2 \ imes 5 = 10\n]\n[\n(20 - 1) \ imes 3 = 19 \ imes 3 = 57\n]\n[\n10 + 57 = 67\n]", "3. Multiply\n[\nS_{20} = 10 \ imes 67 = 670\n]", "---", "Final Result", "The sum of the first 20 terms of the arithmetic sequence with first term 5 and common difference 3 is:", "[\n\boxed{670}\n]", "---", "Why Knowing This Formula Matters (SEO Keyword: arithmetic sequence sum formula)", "Understanding and applying this formula efficiently saves time and reduces errors—especially important in educational content and technical writing. The sum formula allows quick computations crucial for beginners learning sequences and for professionals solving complex modeling problems.", "Key SEO Keywords to Target:\n- Arithmetic sequence sum formula\n- Sum of first n terms of arithmetic sequence\n- Arithmetic sequence calculation tutorial\n- How to find arithmetic series sum\n- Math problem-solving tips for students", "---", "Conclusion", "Finding the sum of 20 terms in an arithmetic sequence becomes intuitive once you master the sum formula. With just a few steps—identify (a_1), (d), and (n)—and plug them into the calculation, even complex sequences feel manageable. Whether for homework, exams, or real-world applications, knowing how to calculate such sums strengthens mathematical fluency and enhances SEO performance in educational content.", "---", "Quick Recap:\n- Formula: ( S_n = \frac{n}{2} [2a_1 + (n - 1)d] )\n- Inputs: ( a_1 = 5 ), ( d = 3 ), ( n = 20 )\n- Result: ( \boxed{670} )", "---", "Written for educators, students, and SEO enthusiasts—this optimized guide ensures your content ranks well while teaching one of math’s most essential concepts!"]
